Civilians desperate to flee Fallujah were having to dodge sniper fire from the Islamic State group, which was keen to keep its "human shields" inside the city as Iraqi forces closed in Monday.

Families who managed to escape told of how IS jihadists opened fire on them as they crossed the Euphrates River on boats and makeshift rafts.

"We know from witness testimonies that civilians... are being forced to stay and are being threatened," Nasr Muflahi, the Norwegian Refugee Council's Iraq director, told AFP.

Footage carried by Iraqi channels showed civilians paddling for their lives on the river, others drifting in inflated wheel chambers.

"People are using anything that floats, from wardrobes to plastic containers," said Caroline Gluck, spokeswoman for the UN's Refugee Agency (UNHCR) in Iraq.

"We know that people have drowned, at least one person was shot by a sniper as he was on some kind of boat or dinghy," she said.

In Amriyat al-Fallujah, where NRC runs the camps housing most of those who have managed to escape IS-held areas, there are new arrivals every day.

"Daesh (IS) shot at us when we left the city from the south. We could hear bullets zipping above our heads as we were crawling through the countryside," said a 60-year-old woman who was too scared to give her name.

"I shouted at them that I would never go back. 'Kill me now', I said. What point is there in living if my children are suffering?," she told AFP on the phone.

The vast majority of the approximately 18,000 people who have fled IS rule since Iraqi forces launched an operation to retake Fallujah two weeks ago are from outlying areas.

The NRC and other groups put the number of civilians trapped inside the city proper at 50,000. For them, leaving Fallujah is almost impossible.

"They put a car bomb on the old bridge and tell people who want to leave the city that they will blow it up against them," said Abu Mohammed al-Dulaimi, a father of six who still lives in the city centre.

He said IS militants had sawed off the top of the car and covered it to make it less visible to enemy aircraft.

"They are under pressure... we can see it in their behaviour. I would say there is one percent of support from the population, it comes from people who get benefits from them," he said.

- Keeping the poor -

Dulaimi said residents saw IS members shaving their beards Sunday for the first time, as they have done in the past to try to mingle with fleeing civilians when they are about to lose a town to the Iraqi forces.

"Daesh has been helping their own families to escape but they are trying to keep the poor people like us," said a 25-year-old woman who reached a camp in Amriyat al-Fallujah after fleeing from Azkrakiyah, a rural area just west of Fallujah.

IS fighters have been moving forces along the dense maze of trails and canals flanking the river, often taking civilians with them as cover.

"There's a lot of internal movement. In Garma, they forced families to move back inside Fallujah," said the UNHCR's Gluck, referring to an area northeast of Fallujah that Iraqi forces recently liberated.

"They did the same thing in Zoba and Saqlawiya," she said. "We've heard this consistently from families."

Two weeks into the offensive, elite forces are struggling to break into the city centre.

The Iraqi government has cited the risks faced by civilians used as human shields in Fallujah as the main factor slowing operations to retake the city.

But Abu Mahdi al-Mohandis, the powerful military commander of the Hashed al-Shaabi paramilitary organisation, argued the plight of trapped families was all the more reason to ramp up the offensive.

"Now Daesh is shooting at women and children to prevent them from leaving," he told reporters on Sunday.

"It's clear that this city is hostage to these gangs, we are not going to just stand here with our hands tied."