The United States has sent more than 51,000 asylum-seekers to wait in dangerous border towns in Mexico as it advises its own citizens not to travel to those regions because of the severe threat of kidnapping, murder and violent crime.

Advocates have been warning about the dangers of Remain in Mexico, or Migration Protection Protocols (MPP), since the program was announced in January. But their warnings have grown louder this week after a new report by Human Rights First revealed that there were at least 340 reports of rape, kidnapping, torture and other violent attacks against people returned to Mexico while they wait for their case to be heard in US immigration court.

Ursela Ojeda, a policy adviser at the Women’s Refugee Commission, has visited the border multiple times to see how the policy is being implemented and said the new report was the “tip of the iceberg”.

“When you see people not showing up for their court hearing in Remain in Mexico, you have to wonder what happened to the people who aren’t there,” Ojeda said.

“There is no way to know why they just missed court – they could have been kidnapped, they could have been killed, they could have been put on a bus by the Mexican government and shoved to another part of the country with no way to get back.”

The Human Rights First report surveys gruesome incidents, such as when a three-year-old boy from Honduras and his parents were kidnapped after being returned to Nuevo Laredo. The mother said the last time she saw her husband he was lying on the ground, beaten and bleeding and told her: “Love, they’re going to kill us.” The kidnappers released the three-year-old and his mother, who doesn’t know if her husband is alive.

A Cuban asylum seeker told the group he saw a group of men stop a taxi outside a Mexican government immigration office and kidnap the four Venezuelan women and girl inside who were being sent to a shelter.

Migrants, mostly from Mexico, are pictured sitting on the ground waiting near the Paso del Norte Bridge at the Mexico-US border, in Ciudad Juarez, Mexico, on 12 September. Photograph: Paul Ratje/AFP/Getty Images

Nuevo Laredo and Matamoros, two of the cities in the Tamaulipas state people are being returned to, are among the most dangerous in the world. The US State department issued a level 4 travel warning for the region because “violent crime, such as murder, armed robbery, carjacking, kidnapping, extortion and sexual assault is common”.

Speaking at the White House on Tuesday, the acting head of US Customs and Border Patrol (CBP), Mark Morgan, ignored multiple questions about what the US government was doing to address the violence facing people sent back to Mexico.

“We’re trying to overcome the message that the cartels have been putting out there that it’s going to be a free ride into the United States,” Morgan said. “We’re now sending the message that, if you’re coming here as an economic migrant, you’re not going to be allowed into the United States.”

He celebrated the program for keeping people out of the US, where they would have been detained or released while they waited for their court date. He also said the program was stopping smugglers and improving due process – though advocates say it is doing the exact opposite.

Shelters and other aid groups are overwhelmed by the migrants pouring into border towns and many are left to sleep and fend for themselves on the streets, without healthcare or work opportunities.

Attorneys say it is nearly impossible to provide legal counsel. Some of the US-based attorneys who have crossed the border have received credible threats of violence and the US has not secured an agreement with Mexico to ensure US attorneys don’t get arrested for practicing law in Mexico without a license.

At the end of August only 34 out of 9,702 people placed into the Remain in Mexico program had legal representation – just 0.4%, according to researchers at Syracuse University’s Transactional Records Access Clearinghouse (Trac).

There is also little accountability for the government’s claim that vulnerable people are exempt from the program on a case-by-case basis. Human Rights First said the screening process is a “farce” and advocacy groups have seen vulnerable groups, including pregnant women and LGBT people, returned.

Democratic 2020 presidential candidate, Julián Castro, on Monday crossed the border with eight gay and lesbian asylum seekers from Cuba, Guatemala and Honduras and a deaf Salvadoran woman and three of her relatives.

“Hours after we were told LGBT and disabled asylum seekers would have their cases heard, they have been returned to Mexico,” Castro said in a tweet. “By law, these migrants are supposed to be exempt from the Remain in Mexico policy – but CBP had decided to ignore their due process. Outrageous.”

In September, a Salvadoran woman who was eight-and-a-half months pregnant and experiencing contractions was apprehended by US border patrol, given medicine to stop contractions in a hospital, then returned to Mexico.

In March, a 27-year-old with the cognitive age of a four-year-old child, was separated from the cousin and son he traveled with and sent back to Mexico. He was reunited with his mother in the US at the end of August after the Guardian reported on his case.

This policy is colliding with other policies that have crippled the asylum system, including a ban on migrants seeking asylum at the border before seeking protection in another country.

On Monday, the Women’s Refugee Commission and other advocacy groups sent a letter urging Congress to investigate the Remain in Mexico program’s “grave human rights and due process violations”.

Advocacy groups also filed a lawsuit against it in February. The policy was blocked in April, but an appeals court temporarily allowed it to continue while the ruling is appealed.

In the court case, the union which represents 2,500 employees in the DHS agency which interviews and adjudicates asylum claims, US Customs and Immigration Services, filed a brief describing Remain in Mexico as “entirely unnecessary” because the system could handle the increase in asylum claims.