Vital relief is still failing to reach thousands of Iraqis displaced by the government's offensive to retake Fallujah from the Islamic State group, drawing severe criticism from some aid organizations.

Some groups said that the fact that families were stranded with no shelter and others crammed in camps with no latrines could have been avoided and was the result of a collective failure.

Elite Iraqi forces on Sunday retook the Islamic State group's last bastion in Fallujah, securing full control of the city after a month-long operation, a spokesman said.

Under fire and nearing starvation, thousands of Iraqi civilians trapped in the jihadist bastion of Fallujah face a "humanitarian disaster", a prominent Norwegian NGO said Thursday.

The Islamic State group battled Iraqi forces and held civilians hostage Wednesday to defend its bastion of Fallujah, where three weeks of fighting has forced tens of thousands from their homes.

The Iraqi army has established a safe corridor that has allowed 4,000 Fallujah residents to flee the jihadist-held city in 24 hours, the Norwegian Refugee Council said on Sunday.

Iraqi government forces fighting to retake Fallujah from the Islamic State group seized a southern neighborhood of the city on Wednesday, officials said.

Led by the country's counter-terrorism service, Iraqi forces and allied militias backed by air support from a U.S.-led coalition began an operation on May 23 to wrest the city from IS.

Militants appeared to have captured at least five Iraqi soldiers near Fallujah Sunday, according to witnesses and online videos, as 13 people were killed elsewhere amid a nationwide surge in unrest.

The latest deaths push to more than 850 the number of people killed this month in Iraq, where authorities are grappling with a deadly standoff in Anbar province and a surge in nationwide bloodshed.