Mr. Alvarado said he fled Honduras last fall after exposing corruption in his hometown and was followed into Mexico by an armed group. Once he reached the United States, he believed he would be safe — even if that meant being detained while waiting for asylum.

“I prefer 1,000 times being jailed,” he said last week at a shelter south of the United States border, “than being dead.”

Instead, as part of a newly expanded Trump administration policy, Mr. Alvarado, 29, a father of two, was sent back to Mexico. He has been waiting for weeks to be summoned for an asylum hearing in California.

Hundreds of asylum seekers are expected to be blocked from waiting in the United States each day under the so-called Remain in Mexico policy, which American officials describe as a deterrent to illegal immigration.

Officials at the Department of Homeland Security said the policy also aims to ease overcrowded detention facilities as they grapple with a recent surge in migrants along the southwestern border. The policy is being challenged in federal court.

“The crisis at our border is worsening, and D.H.S. will do everything in its power to end it,” Kirstjen Nielsen, the homeland security secretary, said this week.

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On Thursday, an official from Mexico’s Foreign Ministry said it was unclear how many asylum seekers might be turned away from the United States under the policy, which he said the Trump administration had expanded without its consultation. The official spoke on the condition of anonymity to discuss a delicate diplomatic issue.

About 633 Central American asylum seekers have been turned away since January, unable to prove sufficient fear of being tortured and persecuted in Mexico. It is a shift from earlier guidelines, which allowed migrants who convinced the authorities that they had a credible fear of returning to their home nations to remain in the United States while their asylum cases were being considered.

Stories of fleeing violence, extortion and corruption in their home countries do not meet the new standard for entry. Many migrants are also unable to obtain lawyers to represent them in court without first meeting them in the United States.

Mr. Alvarado said he was considering sneaking into the United States if his asylum claim was further delayed.

“I’ll have to do it illegally,” Mr. Alvarado said, holding photographs of the men who he said pursued him from Honduras, through Guatemala and into Tapachula, Mexico. “I’ll have to give up everything.”

For the most part, the policy has been rolled out slowly and quietly.

When it began at the San Ysidro port of entry in California in late January, only men traveling by themselves from Central America were told to wait in Mexico as their asylum cases wound through the American legal system. The policy has since expanded to stop entire families from waiting in the United States, although unaccompanied children and Mexican citizens will be allowed to enter.

It is now being enforced at border ports at Calexico, Calif., where President Trump traveled on Friday to tour the border, and El Paso. Ms. Nielsen has directed her department to expand the policy to other legal crossing points from Mexico.

The number of border crossings are nowhere as high as in the early 2000s, when as many as 220,000 migrants crossed the border in a month. Ms. Nielsen estimated last month that border officials had stopped as many as 100,000 migrants in March, up from 76,000 in February.

A State Department report released last month acknowledged the possibility that the migrants were no safer in Mexico from the same gangs that threatened them in Central America than they had been at home.

That is constantly on the mind of Daniel Nuñez, who was working as a security guard in Honduras when gang members opened fire in October, severely injuring several of his colleagues.

Mr. Nuñez fled to the United States border, where he asked for asylum in Calexico, but he was recently sent back to the Mexican city of Mexicali to wait for his immigration hearing.

The closest American immigration court is 30 minutes away. But Mr. Nuñez was told he had to report for his hearing at the San Ysidro port, a nearly three-hour drive.

He has no car, is sleeping in a shelter with about 370 other people and is trying to figure out how to get to San Ysidro. “I was thinking about that,” he said last week, scratching his head. “How I was going to manage.”

In a lawsuit filed in February in federal court in San Francisco, the American Civil Liberties Union accused the government of violating immigration law by returning asylum seekers to Mexico. The Trump administration has maintained it has broad discretion over removal proceedings.

Jacqueline Brown Scott, a lawyer, represents one of the plaintiffs in the case, who is identified in court papers only as Howard Doe for protection. He claims to have fled a drug cartel in Honduras, only to be kidnapped by another cartel in Mexico. He escaped after 15 days and he went to the United States border to seek asylum.

Immigration authorities diverted him to Tijuana. Last week, he appeared in immigration court in San Diego, where Ms. Scott argued that he had a fear of persecution in Mexico. He was sent back again.

“I told them everything, but they didn’t seem to care,” the migrant said in a text message that was viewed by The New York Times, which is withholding his name for security concerns.

Some migrants face additional delays from immigration judges who insist they find legal counsel before their cases can move forward.

Denis Rostran, of Honduras, said he called 10 lawyers who were listed on a document that Department of Homeland Security officials gave to him when he was sent back to Tijuana. None answered. Mr. Rostran said he has slept some nights on the street and has been robbed twice.

A Department of Homeland Security official said it was committed to ensuring migrants have legal assistance at no cost to the government. The agency also is not forcing migrants to return to Mexico who would “more likely than not” be persecuted or tortured there, the official said.

Twenty shelters and churches in Tijuana are housing around 3,000 migrants — and have almost reached their limit, said Esmeralda Siu, the executive coordinator of the Migrant Defense Coalition, in the Mexican state of Baja California. Many of the migrants are awaiting their court dates, she said, and do not expect to leave soon.

That means that newly arrived and future migrants will have few, if any, options for shelter.

“There’s no city, no state, that has sufficient resources to prepare for this,” said Gustavo Magallanes Cortés, the director of migration affairs for Baja California. “They’ve allowed these people to return, but cut the resources for migrants, which has led to chaos. Every day a shelter calls me and says, ‘I’ve run out of food.’”

But many of the waiting migrants are determined to keep hope.

“Many people say they’re doing this just so we get tired and give up on our cases,” said Daniela Diaz, 19, who said she left El Salvador after a member of the MS-13 gang threatened to rape and kill her. She has been living in a shelter in Tijuana since late January.

She is frustrated that following the rules and asking for asylum at a legal port of entry — instead of sneaking into the United States — have resulted in a system of immigration purgatory.

“There’s a lot of people throwing themselves over the wall,” Ms. Diaz said, “and we’re doing this the fair way.”

Azam Ahmed contributed reporting from Mexico City.

A version of this article appears in print on April 5, 2019, on Page A12 of the New York edition with the headline: Anxious Wait in Mexico For Migrants Who Hope U.S. Can Be Safe Haven.

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