A trip to the sauna. A golf club membership. Luxury watches. Neatly packed bricks of red Chinese bills worth $220,000.

The bribes lined the pockets of health care officials across China. Their purpose: to get public hospitals to buy millions of dollars’ worth of sophisticated medical equipment made by foreign companies like General Electric, Siemens, Philips and Toshiba.

A review of dozens of Chinese court cases and internal corporate documents as well as interviews with company insiders showed how foreign firms have become deeply enmeshed in the corruption pervading China’s health care industry. The New York Times reviewed more than a dozen cases in which employees of G.E., Philips and Siemens testified to bribing meagerly paid public hospital officials. In many other cases, Western companies signed off on deals involving third-party contractors who paid bribes and sought kickbacks. Sometimes, the companies continued to sign off on deals involving contractors who admitted to bribery in court.

In one case filed in 2016, a hospital administrator named Wu Dagong was offered more than a $1 million by two G.E. sales representatives to secure the sale of a CT scanner for $4 million. Mr. Wu also took a bribe — the $220,000 in bricks of bills packed in a suitcase — from a G.E. sales contractor, who walked with Mr. Wu to his car and left the suitcase in the trunk. Mr. Wu was sentenced to 15 years.

Siemens, G.E., Philips and others say hospitals often force them to sell through a series of middlemen, where much of the bribery takes place. They said they comply with Chinese and international laws and terminate employees and contractors who they find out are directly involved with wrongdoing.

“We are committed to integrity, compliance and the rule of law in every country in which we do business,” said Tara DiJulio, a G.E. spokeswoman.

China’s nearly 1.4 billion people ultimately bear the cost of the corruption. Salespeople inflated prices for equipment to fund bribes and kickbacks, the Chinese court documents show. They also refused to undercut one another’s pricing, the documents showed, inflating prices for hospitals by 50 percent or even more.

As China grew richer, its people began to yearn for sophisticated smartphones, fresher food and better health care, creating vast business opportunities for Western companies. But China’s growth often outstripped its ability to identify corruption or enforce its own laws. In some industries, like health care, that has created a culture of bribery and corruption.

China is expanding its social safety net and faces pressure to treat an aging population growing increasingly susceptible to chronic diseases. China will spend $1 trillion on health care annually by 2020, according to a McKinsey report.

Foreign companies like G.E. and Siemens dominate the market for CT scanners, M.R.I.s and other equipment China needs to detect cancer and other chronic diseases, though local rivals are catching up. Last year, China imported more than $22 billion worth of medical devices.

“Corruption is endemic in the health care sector in China,” said Yanzhong Huang, a senior fellow for global health at the Council on Foreign Relations. He said one factor was lower salaries for doctors and health care administrators compared with peers in the private sector.

“The direct result,” Mr. Huang added, “is that the patients will pay more.”

Officials in China and elsewhere have tried for years to stamp it out. Siemens promised in 2008 to scrutinize deals for graft, including sales of medical equipment in China. Chinese officials five years ago ordered GlaxoSmithKline, the British drug maker, to pay nearly $500 million for bribing doctors and hospitals.

Then, last year, an investigation by the German newspaper Suddeutsche Zeitung highlighted dozens of recent Chinese cases in which Siemens employees and sales representatives were accused of bribery. In its own review of court cases involving Siemens, Philips, G.E. and Toshiba, The Times found that one or more layers of middlemen often worked with hospital directors to set prices that included bribes and kickbacks. A review of more than a dozen recent deals from Siemens and G.E. shows that the price was at least 50 percent higher or even double when they involved a third-party distributor, according to hospital and corporate documents.

“Layering a distributor or vendor in between creates more space and creates the possibility for those vehicles to harbor slush funds or falsify documents along the way and funnel bribes,” said Wade Weems, a former United States prosecutor and lawyer for Kobre & Kim, a law firm.

In the cases reviewed by The Times, the Western companies were not accused of wrongdoing, as prosecutors kept their focus on corruption by Chinese officials.

In one case, a Siemens sales representative testified in 2016 to paying nearly $900,000 to a hospital director in the city of Qinzhou to secure the sale of a Siemens M.R.I. machine. The Siemens representative, named only Jin in court documents, shoved stacks of cash into boxes and delivered them to the trunk of the car of the hospital director, Chen Fengkun. Mr. Chen was sentenced to 15 years in prison. Mr. Jin got three years.

The case was covered in detail in a state-owned newspaper called The Legal Evening News, which pointed to 19 other bribery cases involving Siemens employees or contractors from 2014 to 2015.

In another case from 2016, Gao Xuezhong, the president of a hospital in Anhui Province, was convicted of taking bribes from a Siemens sales contractor and from a Siemens sales manager named in court as Mr. An. The bribes included cash as well as homes for Mr. Gao’s wife and for his daughter. At one point, Mr. Gao and Mr. An marked up the price of a $1.3 million M.R.I. machine to $1.7 million. They and intermediaries pocketed the difference.

Mr. Gao was sentenced to 11 years in prison. Mr. An’s name appeared in another bribery case around the same time.

Stefan Schmidt, a spokesman for Siemens Healthineers, the health division of Siemens, said that it opened an investigation into Mr. An but that he refused to cooperate and resigned.

Siemens trains and monitors its third-party sales representatives, Mr. Schmidt said. “Whenever we identify misconduct on the part of distributors,” he said, “we end our relationship with them.”

Some of those firms remain involved anyway.

To complete the Anhui sale, Mr. Gao and Mr. An used an importing firm called Anhui Yameiya Import and Export Trade Company. Yameiya, which is based in Anhui, has been named in several other corruption cases involving equipment made by Siemens and G.E. The company declined to answer questions.

Siemens said that it terminated its relationship with Yameiya when it learned of the cases in 2014. But Yameiya is involved in dozens of recent transactions as a middle agent for Siemens, G.E., Philips and Toshiba equipment, according to publicly available procurement documents from Chinese hospitals. Just a few months ago, Yameiya bought a Siemens M.R.I. machine from an authorized dealer. In another deal a year ago, Yameiya sold a G.E. device to a hospital in Anhui Province.

Siemens, Philips, Toshiba and G.E. said that Chinese law required hospitals to hire third-party companies to import foreign equipment and that they had no say in who was involved. “Being a responsible company, everyone in Philips is expected to always act with integrity,” said Steve Klink, a company spokesman.

Often the bidding is just a show. A court last year convicted Xiao Feng, an administrator at an unidentified Beijing hospital, of taking $330,000 in bribes to buy Toshiba and Siemens equipment. A second administrator identified as Dong in court documents testified that the hospital had already chosen the device and “the bidding is simply a formality that makes the procurement process legal and legitimate.”

Companies know the bids are rigged, said an authorized seller of Toshiba products identified as Han in court documents. “This is an unspoken industrial rule,” Mr. Han testified during Mr. Xiao’s trial. “We cooperate with each other.”

Canon Medical Systems Group, which bought Toshiba’s medical device business in 2016, has “a zero-tolerance policy toward bribery and unethical business practices,” said Hiroko Eno, a spokeswoman.

That cooperation can lead to higher prices for hospitals, say health care and compliance experts.

Public bidding documents indicate device prices differ sharply between deals with third-party brokers and direct sales. When Siemens won a bid to sell an M.R.I. machine to the Chinese Academy of Medical Sciences in Beijing in 2016, the price was $2,800. In another deal, in which Siemens sold the M.R.I. machine through a third-party broker called Chongqing Kangtian Medical Equipment, the price was $4,700.

A woman who picked up the phone at Kangtian and declined to give her name said the deal went through normal bidding processes. She declined to answer further questions or refer them to executives.

Using a third-party sales force does not insulate a foreign company from allegations of wrongdoing. Chinese law requires the companies to sign off on the deals through an authorization letter. Under the Foreign Corrupt Practices Act in the United States, companies can be held responsible for the conduct of lower-level distributors.

“If there is a circumstance under which the difference between a direct sale and third-party sale is double, that’s a red flag,” said Richard Bistrong, who went to prison for bribing foreign officials and now advises companies trying to beef up compliance.

Mr. Schmidt, of Siemens, said that the publicized price for a third-party deal “does not always tell the whole story because additional contract components contribute to the price,” including services and supplies.

Meng-Lin Liu, the former chief compliance officer for Siemens’s health care unit in China, said device makers were well aware of the bribery. “It’s a conspiracy scheme,” said Mr. Liu, who said he was fired for speaking out about corruption in the China subsidiary. He lost a whistle-blower retaliation lawsuit in New York against Siemens in 2013.

“Everyone knows. The problem is how to make people speak up. Because everyone lives on this system, no one dares to report others,” added Mr. Liu, who also goes by Louis Liu.

Siemens said Mr. Liu’s allegations were investigated and “found to have no merit.”

The Chinese market is shifting. Beijing hopes to increase the number of domestically produced devices in public hospitals by half by next year. The trade war with the United States could also make American equipment less competitive.

Still, as long as China remains a fast-growing market, say experts, a culture of graft could linger among China’s hospitals.

“These companies think they’ve got their nose clean because the corruption is being done by an agent in the middle,” said Peter Humphrey, a British private investigator who was detained in China while working for GlaxoSmithKline during its bribery investigation.

“I’ve seen so much of this nonsense,” he added. “This is all window dressing.”