BAGHDAD » Iraqi special forces swept into Fallujah today, recapturing most of the city as the Islamic State group’s grip crumbled after weeks of fighting. Thousands of trapped residents took advantage of the militants’ retreat to flee, some swimming across the Euphrates River to safety.

Residents described harrowing escapes even after IS fighters abandoned some checkpoints that had them bottled up in the city. On the river, some boats packed with people overturned in the water. Others picked their way down roads laced with hidden bombs that killed several. In some cases, IS allowed people to leave only if they took the jihadis’ families with them.

After weeks of heavy battles since the offensive began in late May, it appeared that IS defenses in much of the city collapsed abruptly.

In the early morning today, Iraqi forces punched into the city center, meeting intense fighting. But by evening, the special forces commander Brig. Haider al-Obedi told the Associated Press that his troops controlled 80 percent of the city, with IS fighters now concentrated in four districts on its northern edge.

It was a major step toward regaining the Islamic State group’s last major foothold in Iraq’s western Anbar province, the heartland of the country’s Sunni minority. The militants overran the city in early 2014, the first urban area to fall into its hands before it overran most of Anbar and much of northern Iraq.

Over the past year, Iraqi forces backed by U.S-led airstrikes have city-by-city regained large parts of that territory — though the biggest prize, Iraq’s second largest city, Mosul, and surrounding territory in the north, remains in IS control, liked to its holdings in neighboring Syria.

This evening, Iraqi Prime Minister Haider al-Abadi spoke on national TV from the joint command center, congratulating the troops on their victories. “We promised to liberate Fallujah, and it has returned to the embrace of the nation,” he said.

Iraqi forces have “tightened their control over the inside of the city, and there are some pockets that need to be cleaned out within hours,” he said.

In the early hours, special forces pushed into Fallujah’s central al-Nazzal district, which had served as a base for the militants with weapons warehouses and command centers, al-Obeidi said. Backed with air support from the US-led coalition and Iraqi air force, the troops were able to move into the center at around 6 a.m. They seized the main government complex, which includes municipality offices that IS had torched, the police station and other government buildings.

“Iraqi forces are now in the center of the city. They had not been there since the beginning of 2014,” al-Obeidi said.

IS fighters were still holding out in the nearby central hospital, al-Obeidi said. Throughout the day and into the night, Iraqi forces surrounded the hospital, clashing with snipers on adjacent buildings. But they were holding back from storming the building, fearing there were patients inside that the militants would use as human shields, he said.

Meanwhile, troops were clearing roadside bombs from recaptures areas, including the government complex and the highway west of the city, linking it to Baghdad, al-Obeidi said.

Aid groups had estimated that 50,000 civilians had been trapped inside Fallujah when the assault began several weeks ago, and they say that 30,000 to 42,000 of those had fled since then. They have largely been staying in camps in areas around the city.

The Norwegian Refugee Council said the thousands more people fleeing the city were overwhelming services at the camps, with many sleeping in the open and drinking water in short supply.

The group cited a 69-year-old Fallujah resident saying IS fighters suddenly disappeared from many streets Thursday evening, as neighbors saw them evacuating checkpoints and driving away in vehicles loaded with food and fuel. The news prompted many residents to prepare to escape.

One resident, Ali al-Mohammadi, told the AP he fled today with more than a dozen other relatives, including several children, but ran into IS fighters deployed at the banks of the Euphrates, which runs along the western edge of Fallujah.

The militants beat them and fired shots in the air to drive them back, but finally as a crowd grew, the fighters relented and began allowing them to cross in small boats.

The 29-year-old al-Mohammadi said that meanwhile he went to another part of the river and swam across to safety along with others. As he swam, he saw two boats capsize, spilling passengers into the water. They seemed to all make it to land, some using inner tubes they had brought with them, he said.

Others tried to flee down a road leading out of the city to the south, only to find it mined with explosives. Mohammed Ismail, a 32-year-old trying to escape with his family, said militants on the road fired in the air to stop them.

“They forced us to stay until they could bring out the families and children of Daesh to come with us,” he said, using the Arabic acronym for the Islamic State group. “The price of our leaving was to bring their families with us.”

In the pre-dawn darkness, an IS fighter led them down a road past the explosives. Still, Ismail said, he saw one mine blow up, killing at least two people, before the crowd made it to Iraq military-controlled territory.

There, Iraqi troops separated women and children from the young men, who were then questioned to find any escaping militants.

The conflict in Iraq has forced more than 3.3 million people to flee their homes. Iraq is also hosting up to 300,000 refugees who have fled the civil war in neighboring Syria. Most are living in camps or informal settlements.

Nasr Muflahi, the Norwegian Refugee Council’s Country Director in Iraq, called for more international aid to help those fleeing Fallujah. Services in camps are already overstretched, and more will be needed, he said.

“International donors need to act now,” Muflahi said, “so that we can help Iraqi families who have been through long hellish months of widespread hunger, terror and despair.”

Associated Press correspondent Sarah El Deeb contributed to this report from Beirut.