The bridge connects Del Rio in Texas to Mexico’s Ciudad Acuña and the temporary camp there has grown with staggering speed in recent days.
The mostly Haitian migrants, who have crossed the Rio Grande, are sleeping under the bridge in squalid conditions.
The US government has been facing a surge of migrants at the border.
Earlier this year, it was reported that the number of migrants detained at the US-Mexico border in July exceeded 200,000 for the first time in 21 years, government data shows.
And last month, the authorities arrested more than 195,000 migrants at the Mexican border, according to government data released on Wednesday. This summer’s numbers represent a significant increase from the 51,000 arrested in August 2019.
The makeshift camp has few basic services, and migrants waiting in temperatures of 37C (99F) are said to be going back to Mexico to get supplies.
They are said to be mostly Haitians, with some Cubans, Venezuelans and Nicaraguans also present, reports say.
They appear to be part of a larger wave of Haitians heading north, many of whom arrived in Brazil and other South American nations after the 2010 earthquake, the Washington Post reports.
According to Del Rio Mayor Bruno Lozano, more than 10,500 migrants were under the Del Rio International Bridge as of Thursday evening, Reuters news agency reports.
Ramses Colon, a 41-year-old Afro-Cuban asylum seeker who worked in Peru to save money for the trip, said the Del Rio camp was “chaos”.
“You stand there among thousands with your little ticket waiting for your turn,” he told the Washington Post. Migrants have been given tickets with numbers while they wait to be processed.
Border Patrol said in a statement it was increasing staffing in Del Rio to facilitate a “safe, humane and orderly process”.
“To prevent injuries from heat-related illness, the shaded area underneath Del Rio International Bridge is serving as a temporary staging site while migrants wait to be taken into USBP [US Border Patrol] custody,” it added.
Drinking water, towels and portable toilets have been provided, the statement added.
Republican Congressman Tony Gonzalez, whose district includes Del Rio, said in an interview with Fox News that the situation is “as bad as I’ve ever seen it”.
“When you see the amount of people and how chaotic it is and how there is literally no border, folks are coming to and from Mexico with ease, it’s gut wrenching and it’s dangerous,” Mr Gonzalez added.
Immigrants rights activist Jenn Budd told the BBC she believes the situation is a direct result of the Biden administration’s extension of Title 42, a Trump-era pandemic policy that allows the US to swiftly expel undocumented migrants.
“When you have large numbers of people who need refugee or asylum status, and have essentially closed down the system that deals with refugees and asylum seekers, then you have people with no other choice but to cross illegally,” said Ms Budd, a former Border Patrol agent and intelligence analyst.
“In order to have national security, you have to have a robust and humane asylum system,” she added. “Otherwise, people end up spilling over illegally, nobody gets checked, and then people get hurt like this”.
On Thursday, a federal judge blocked the administration from continuing to use title 42. The judge said that the policy does not give the government power to expel migrants or deny them opportunities to seek asylum. The order will go into effect in 14 days, giving the government an opportunity to appeal.
Since taking office, President Joe Biden, who had vowed to reform US immigration, has created a taskforce to reunify migrant children with their families, paused construction of Donald Trump’s border wall, and called for reviews of legal immigration programmes terminated by his predecessor.
Mr Biden has also presided over a record-breaking influx of arrivals to the US southern border, including hundreds of unaccompanied children who are being held in US immigration detention facilities.