A "human catastrophe" is unfolding in the Iraqi city of Fallujah as the Iraqi army's surge was checked by heavy ISIS bombardment.

ISIS fighters fought back on Monday night to halt the advance of elite Iraqi soldiers in a southern district of the city. Thousands of civilians remain trapped, prevented from escape by road side bombs.

"Our forces came under heavy fire, they are well dug in in trenches and tunnels,'' said an army commander in Camp Tariq, an army base south of Fallujah.

The city, which lies 50km to the west of Baghdad, has been under siege for more than six months with aid organisations barred from the centre.

A staff member of Fallujah's main hospital said they received reports of 32 civilians killed in Monday's fighting. Aid organisations have warned more than 50,000 civilians have limited access to water, food and health care.

"A human catastrophe is unfolding in Fallujah. Families are caught in the crossfire with no safe way out," said Jan Egeland, secretary general of the Norwegian Refugee Council, one of the organisations helping families displaced from the city.

"For nine days we have heard of only one single family managing to escape from inside the town,'' he said in a statement on Tuesday. "Warring parties must guarantee civilians safe exit now, before it's too late and more lives are lost."

Fallujah is the second-largest Iraqi city still under control of ISIS militants, after Mosul, their de facto capital in the north that had a pre-war population of about two million. It is seen as the bastion of Sunni Iraq and fought against both the US occupation and the Shi'ite-led Baghdad government that replaced Saddam Hussein in 2003.

It was the first city to fall to ISIS' control in January 2014 and would be the third major city recaptured by the government in recent months if the offensive succeeds. Capturing it would give the government control of the major population centres of the Euphrates River valley west of the capital for the first time in more than two years.

Prime Minister Haider al-Abadi announced the assault on Fallujah on May 22 after a spate of bombings that killed more than 150 people in one week in Baghdad, the worst death toll so far this year. A series of bombings claimed by Islamic State also hit Baghdad on Monday, killing more than 20 people.

Additional reporting by Reuters.