At least two children drowned with their mother while trying to flee the Islamic State stronghold of Fallujah on Saturday. A man also drowned.

Nine others were missing as civilians continued their attempts to cross the Euphrates river to escape Fallujah, which is currently the target of a major offensive against ISIS.

The father of the two children, Abu Tabarak, confirmed over the phone that he had witnessed their boat sinking, killing them and his wife.

"I've seen with my own eyes my family disappear under the water," he said, according to Reuters.

"There was no place for me on the departing boat, so I had waited with my second daughter for the next one," he added. "Life has become worse than hell in Fallujah; hunger, death and shelling. And after all that suffering, I lost my family before my eyes."

Tabarak's family had spent four days picking their way across the city, moving slowly to avoid shelling, sniper fire and explosive devices.

His story highlights the plight of an estimated 50,000 civilians in Fallujah, which lies 50 km from Baghdad. It was the first Iraqi city taken by ISIS in January 2014, as militants seized swathes of territory there and in Syria.

Already suffering shortages of water, food and medicines, the city is now under bombardment from Iraqi forces, allied Shi'ite militias and aircraft from the US-led coalition, as they pursue the battle to retake Fallujah that began on May 23.

A leader of the Iran-backed Shi'ite coalition taking part in the offensive said Fallujah was now all but encircled, the only side that had yet to be secured by pro-Baghdad forces being part of the western bank of the Euphrates.

"We are now at the gates of Fallujah," Abu Mahdi al-Muhandis, deputy leader of the Popular Mobilization Force, told a televised news conference.

Civilians have been using anything that floats to help them cross the river, provincial council head Shakir al-Essawi said.

"They are using empty refrigerators, wooden cupboards and kerosene barrels as makeshift boats," Essawi told Reuters. He said more than 1,000 families had managed to get across.

"It's totally unsafe and this is why innocent people are drowning."

A doctor at the al-Amariya hospital, west of Baghdad, told The New Arab that 18 bodies in total were pulled from the Euphrates river on Saturday, including seven children and three women.

"They tried to escape from death but were swallowed up by the river," Dr Mohammad al-Isawi said, adding: "Daesh is responsible for these deaths and whoever claimed the existence of safe passage out of Fallujah."

Some die before even reaching the river, killed by sniper fire from Islamic State lines or by explosive devices along the roads, said Jassim Alwan, a police captain in Ameriyat Fallujah. Last week, the situation was branded a "human catastrophe" by Jan Egeland, secretary general of the Norwegian Refugee Council which is helping families displaced from the city.

Prime Minister Haider al-Abadi said on June 1 the offensive on the city had been slowed down in order to protect civilians.

Additional reporting by Reuters.