WASHINGTON — Rushing to help Afghans who face retribution for working alongside American troops in their home country, the House voted overwhelmingly on Tuesday to speed up the process that would allow them to immigrate to the United States.

With the American military in the final phases of withdrawing from Afghanistan after 20 years of war, more than 18,000 Afghans who have worked for the United States as interpreters, drivers, engineers, security guards and embassy clerks are stuck in a bureaucratic morass after applying for Special Immigrant Visas, available to people who face threats because of work for the U.S. government.

“I can say with confidence that I might not be here today had it not been for these men and women,” said Representative Jason Crow, Democrat of Colorado and a former Army Ranger who is the lead sponsor of the bill.

The measure, passed 366 to 46, would waive a requirement for applicants to undergo medical examinations in Afghanistan before qualifying, instead allowing them to do so after entering the United States. The first in a series of bipartisan bills intended to smooth the visa process, it aims to shorten the long waiting period, which can be as long as six or seven years for some applicants.

Mr. Crow said waiving the medical examination requirement would save the average applicant about a month on processing the visa. The bill requires that applicants complete their examinations within 30 days of arriving in the United States.

“In combat and in a war zone, every hour matters,” Mr. Crow said. “A month will save many, many lives.”

Some of the “Afghan allies” awaiting visas have spoken out about the threats they face from the Taliban.

Since 2014, the nonprofit No One Left Behind has tracked the killings of more than 300 translators or their family members, many of whom died while waiting for their visas to be processed, according to James Miervaldis, the group’s chairman and an Army Reserve noncommissioned officer.

“It is a life and death situation,” said Representative Brad Wenstrup, Republican of Ohio. “It’ll be a black eye on the United States if we don’t do everything in our power to protect these allies.”

Both Mr. Crow and Representative Cliff Bentz, Republican of Oregon, said staff members had worked tirelessly over the past few weeks to alleviate the concerns of some House Republicans, who worried that eliminating the medical examination requirement could lead to the spread of disease once the applicants entered the United States. They added the 30-day deadline to address those fears.

The measure is one of several being pushed by a bipartisan coalition of lawmakers, many of them military members or veterans who have worked with translators, drivers and fixers in Afghanistan and other combat zones.

The group has also pressured the Biden administration to carry out a mass evacuation of Afghans who are awaiting their visas, an idea that President Biden embraced last week, saying, “Those who helped us are not going to be left behind.”

But first, the Afghans must qualify for visas. Only one clinic in the country — a German facility in Kabul — does the examinations, requiring some people to travel long distances through dangerous conditions. And the exams are expensive, Mr. Crow said.

A separate bill introduced by the group would expand the universe of eligible Afghans by removing what its proponents call “burdensome” application requirements, including a “credible sworn statement” of a specific threat and proof of a “sensitive and trusted” job. Instead, it would in effect stipulate that any Afghan who helped the U.S. government by definition faced retribution and should be able to apply for a visa. The bill would also increase the number of visas available.

Biden administration officials have said they plan to relocate the Afghan allies outside Afghanistan, possibly to Guam, to await the processing of their visa requests.