Seventy members of the migrant caravan that reached the United States' southern border last week crossed into the U.S. Thursday morning, turning themselves in to American authorities in order to claim asylum, organizers said.

Those 70 brought the total number of migrant caravan members who have crossed into the U.S. in order to claim asylum to 158, according to Alex Mensing, a project coordinator for Pueblo Sin Fronteras, the group that organized the caravan.

Members of a caravan of migrants from Central America who traveled through Mexico camp at Tijuana's pedestrian west border crossing with the U.S., April 30, 2018. (MEGHAN DHALIWAL/The New York Times via Redux Pictures) Members of a caravan of migrants from Central America who traveled through Mexico camp at Tijuana's pedestrian west border crossing with the U.S., April 30, 2018.

All have crossed at the San Ysidro Port of Entry between San Diego and Tijuana, Mexico, including the 70 who turned themselves in at 9 a.m. Thursday. They were the largest group to be accepted for processing so far, organizers said.

There are around 70 members of the caravan still in Tijuana waiting to cross into the U.S. and claim asylum, Mensing said.

Members of a caravan of migrants from Central America enter the United States border and customs facility near Tijuana, Mexico, where they are expected to apply for asylum, May 2, 2018. (Edgard Garrido/Reuters) Members of a caravan of migrants from Central America enter the United States border and customs facility near Tijuana, Mexico, where they are expected to apply for asylum, May 2, 2018.

U.S. Attorney General Jeff Sessions said on Tuesday that he was rirecting 35 federal prosecutors and 18 immigration judges to the southwest U.S. border with Mexico to assist with the caravan and immigration issues there.

Since the migrants first tried to gain entry to the U.S. side of the San Ysidro border crossing on Sunday, authorities have at times blocked their access, saying the facility had reached capacity.

A child traveling with a caravan of migrants from Central America sits at a camp near the San Ysidro checkpoint, after U.S. border authorities allowed the first small group of women and children entry from Mexico overnight,May 1, 2018. (Edgard Garrido/Reuters) A child traveling with a caravan of migrants from Central America sits at a camp near the San Ysidro checkpoint, after U.S. border authorities allowed the first small group of women and children entry from Mexico overnight,May 1, 2018.

Members of the caravan, composed largely of women and children mainly from Honduras, had woven their way through Mexico since late March. The group at one point last month numbered over 1,000, but its size diminished significantly after it gained the frequent ire of President Donald Trump and the subsequent assistance of Mexican authorities.

Members of the caravan, which also includes people from Guatemala and El Salvador, say they are fleeing violence in Central America and fear returning home.

ABC News' Mike Levine contributed reporting from Washington.