BAGHDAD — After weeks of battling the Islamic State, Iraqi forces quickly entered central areas of Fallujah on Friday, as thousands of civilians fled in a new wave of displacement that has overwhelmed the ability of aid agencies to care for them.

Reporting little resistance from Islamic State fighters, counterterrorism forces raised the Iraqi flag over the main government building in central Fallujah, officers and state television reports said. They said that progovernment forces moved on to besiege the city’s main hospital, which was the first target of US forces when they invaded the city in 2004 and in recent months has served as a headquarters complex for the Islamic State.

The rapid and unexpected gains suggested a shift in tactics by the Sunni extremists of the Islamic State, also known as ISIS or ISIL, or perhaps a sign of their weakness, as they abandoned their dug-in positions and regrouped in western neighborhoods of Fallujah. That allowed thousands of civilians to flee across two bridges over the Euphrates River beginning on Thursday. Aid groups had said the civilians were being held as human shields.

Though the battle appeared far from over, Iraqi commanders on the ground were optimistic that the advance — which had slowed in the face of Islamic State snipers, roadside bombs, and tunnel networks that allowed fighters to move around undetected — would continue.

“ISIS has lost its power to defend Fallujah,’’ Colonel Jamal Lateef, a police commander in Anbar province, said. “Its defensive lines have collapsed, and the battle of Fallujah will be over in no time.’’

Lieutenant General Adbulwahab al-Saadi, a commander of Iraq’s counterterrorism forces who is in charge of the Fallujah operation, said in a brief telephone interview that “ISIS has collapsed in Fallujah very fast,’’ and he said his forces were moving to northern and western neighborhoods.

The United States, which has led a coalition targeting ISIS with airstrikes for almost two years in Iraq, has supported the battle for Fallujah with air power, even as it has raised concerns about the role of Shi’ite militias backed by Iran in the fight. It has expressed fear that the participation of Shi’ite forces in assaulting a Sunni city like Fallujah would heighten sectarian tensions.

Colonel Christopher C. Garver, a US military spokesman in Baghdad, said coalition airstrikes on Friday aided the taking of the government building in Fallujah by knocking out two heavy machine guns nearby.