SAN DIEGO (AP) — A new policy to deny asylum to anyone who shows up on the Mexican border after traveling through another country threatens to exacerbate overcrowding at severely strained U.S. immigration detention centers and makeshift holding areas.

Photos and video of Vice President Mike Pence's visit Friday to McAllen, Texas, showing men crammed behind chain-link fences offered the latest glimpse into squalid conditions at Customs and Border Protection facilities. Women are being held in smaller tents at the station.

The Border Patrol housed 900 people in an area with capacity for 125 in E...