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.

Tens of thousands of others who have managed to flee the city as Iraqi forces press a bid to dislodge the Islamic State group also find little relief on their way out.

"We have a humanitarian disaster inside Fallujah and another unfolding disaster in the camps," the Norwegian Refugee Council (NRC) said in a statement.

"Thousands fleeing the cross-fire after months of besiegement and near starvation deserve relief and care, but our relief supplies will soon be exhausted," NRC chief Jan Egeland said.

Fallujah, which lies west of Baghdad, is one of the last two major Iraqi cities IS controls, the other being Mosul.

For three weeks, Iraqi forces backed by a US-led coalition air strikes have alternated barrages of artillery fire with an attempt to move forward in street battles against IS jihadists.

Progress has been hampered however by IS's systematic use of civilians as human shields.

"Make no mistake: There is absolutely nothing safe for civilians fleeing Fallujah. No safe exits, no safe passage, no safe haven without risking their lives," Egeland said.

"They risk being shot at, killed by explosive devices on the roads, or drowning while crossing the river."

According to the International Organization of Migration (IOM), at least 48,000 people have managed to escape since the offensive was launched on May 23.

"Those who flee IS-controlled areas and manage to make it to safety will soon find out there is very little we can offer them: we are running out of food, drinking water and medical services," the NRC said.

The group, which is working in camps for the displaced near Fallujah, said it can only offer survivors three litres (six pints) of water a day -- well under the 10 litres needed in temperatures nearing 50 degrees Celsius (122 degrees Farenheit).

More than three million Iraqis have fled their homes since a lightning IS offensive in 2014 saw swathes of the country fall into jihadist control.