Thousands of civilians are fleeing Fallujah after the city was declared liberated from the Islamic State group, the United Nations said, while an Iraqi commander reported fierce clashes as elite counterterrorism forces pushed to clear out the remaining militants.

IS fighters launched missiles, detonated a suicide car bomb and deployed snipers against Iraqi forces, Brig. Gen. Haider al-Obeidi said. "Iraqi forces are still advancing despite the strong clashes," he said.

Over the past three days, the U.N. says that nearly ten thousand families have fled Fallujah amid the heavy fighting. More than 80,000 civilians have fled Fallujah and its surrounding areas since the operation to retake the city from IS was announced last month, according to the U.N.

"Agencies are scrambling to respond to the rapidly evolving situation and we are bracing ourselves for another large exodus in the next few days as we estimate that thousands more people remain trapped in Fallujah," said Bruno Geddo, the representative for the U.N.'s refugee agency in Iraq, in a statement Sunday.

Iraqi Prime Minister Haider al-Abadi declared victory for Iraq on Friday after special forces recaptured most of the city after weeks of fighting. This leaves Mosul, Iraq's second largest city, as the only remaining urban stronghold for IS militants in the country.

The Norwegian Refugee Council says the civilian exodus has overwhelmed camps run by the Iraqi government and humanitarian groups, leaving thousands without shelter or proper sanitation.

"Right now as we speak there are thousands without any tents, without any shelter, they have slept overnight out in the open," said Karl Schembri with the Norwegian Refugee Council, an international humanitarian organization that does extensive work in Anbar province, which surrounds Fallujah. Schembri said the humanitarian situation in Anbar province following the Fallujah operation is potentially "catastrophic."

The conflict that erupted in Iraq after IS blitzed across the country in the summer of 2014 has forced more than 3.4 million people to flee their homes.