For the third time in seven years, U.S. officials are scrambling to handle a dramatic spike in children crossing the U.S.-Mexico border alone, leading to a massive expansion in emergency facilities to house them as more kids arrive than are being released to close relatives in the United States.

More than 22,000 migrant children were in government custody as of Thursday, with 10,500 sleeping on cots at convention centers, military bases and other large venues likened to hurricane evacuation shelters with little space to play and no privacy. More than 2,500 are being held by border authorities...