ROME — More than 4,000 would-be refugees were rescued at sea Thursday in one of the busiest days of the Mediterranean migrant crisis, and at least 20 died trying to reach Europe as Libyan-based smugglers took advantage of calmer seas to send desperate migrants north.

The death toll was likely to grow far higher, however. The Libyan coast guard also reported two overturned boats between the coastal cities of Sabratha and Zwara. Only four bodies were found, raising fears that the rest of those on board had died.

In this picture series starting at upper right, migrants try to jump in the water before their boat overturns Wednesday off the Libyan coast. The Italian navy later recovered a few bodies. The Associated Press

A woman disembarks with her baby from an Italian navy ship as it arrives at a Sicilian harbor Thursday. Improved weather conditions have led to an increase in the number of migrants risking the crossing from Libya. The Associated Press

Migrants plead for help after their boat overturned off the Libyan coast Thursday with more than 100 passengers on board. Officials said 88 people were rescued and about 20 bodies were spotted in the sea. The Associated Press

Overall, the Italian coast guard said it had coordinated 22 separate rescue operations Thursday that saved more than 4,000 lives.

“That probably is a record,” said coast guard spokesman Cmdr. Cosimo Nicastro, noting that previous highs have been in the range of 5,000 to 6,000 over two days.

One 5-year-old boy got special treatment: He was airlifted from his rescue vessel to the island of Lampedusa, suffering from hypothermia, Nicastro said.

At least one smuggler boat sank off Libya’s coast, and 20 bodies were spotted floating in the sea, said Navy Lt. Rino Gentile, a spokesman for the EU’s Mediterranean mission. Photos tweeted by the mission showed a bright-blue dinghy submerged under the weight of migrants waving their arms in hope of rescue as an EU aircraft flew overhead. None had a life jacket.

Two Italian coast guard ships and the Spanish frigate Reina Sofia responded to the scene. Nicastro said 96 people were rescued.

Barbara Molinario, spokeswoman for the U.N. refugee agency in Italy, said favorable weather conditions from May to October often encourage migrant crossings. She said about 40,000 people have been rescued so far this year, not counting the recent ones, compared with 47,500 over the same period in 2015.

Among those coming ashore Thursday in Sicily were the survivors of a dramatic capsizing a day earlier off Libya’s coast. Footage provided by the Italian navy showed the steel-hulled smuggler ship rocking under the weight of its passengers until it finally flipped, sending migrants into the water or clambering up the side.

The Italian navy vessel Bettica brought the survivors and five bodies ashore in Porto Empedocle, Sicily. Red Cross workers took at least one migrant away in a stretcher, while rescue teams in white hazmat suits carried children down the plank to shore.

In other rescues, a Libyan navy spokesman said a total of 766 migrants were rescued by the Libyan coast guard Thursday.

Col. Ayoub Gassim said two other capsized boats were found empty in waters between the two cities and only four bodies were retrieved, with the rest of those aboard feared dead. He said he had no other details, including how many migrants had been aboard the boats.

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