Thousands more people have fled from the Islamic State-held city of Fallujah through an exit corridor secured by the Iraqi army even as an offensive to retake the city from the terror group rages into its third week.

According to reports from the United Nations and the Norwegian Refugee Council (NRC), 3,300 residents fled Monday, joining another 4,000 who escaped during the weekend. The number of residents fleeing the city now totals almost 30,000 since Saturday, the Associated Press reports.

"We expect thousands more to be able to leave in the coming days,'' Joint Operation Command spokesman Brigadier Gen. Yahya Rasool said.

The development, Rasool said, was made possible after the al-Salam junction route was secured by the Iraqi army. "There were exit routes previously, but this is the first to be completely secured and is relatively safe."

Lise Grande, U.N. humanitarian coordinator for Iraq, said around 50,000 civilians could still be trapped inside Fallujah, located about 40 miles (65 km) west of Baghdad. ISIS militants are reportedly demanding 150,000 Iraqi dinars or around $130 from each person to let them leave. Last week they allegedly shot at a group of people attempting to flee across the Euphrates River.

Many residents left behind in Fallujah are also in urgent need of assistance as the city is running low on humanitarian resources, according to NRC Country Director in Iraq Nasr Muflahi.

Mufhali said the international community has been urged to provide more help as "needs are great and the resources are limited."

U.N. High Commissioner for Human Rights Zeid Ra'ad al-Hussein meanwhile said there are credible eyewitness reports that some men and boys who escaped are facing physical abuse from armed militias operating in support of the Iraqi security forces, reports the Voice of America (VOA).

There are also allegations that some individuals have been summarily executed by armed groups, he added.

Grande said 7,200 men and boys had been separated from their families. About 2,000 are still being held in Fallujah, reports say.

Fallujah has been under ISIS control for over two years now and it is the last major city in western Iraq that is still held by the extremist group.

On Sunday, the Iraqi command announced that key areas to the west of Fallujah have been retaken and that Iraqi forces are pushing deeper into the city from its southern edges.

Authorities said the militants are putting up a tough fight in defending the city, which has long been an insurgent bastion where U.S. forces fought the heaviest battles of their own during the 2003-2011 occupation of Iraq.