Hundreds of people fled the Falluja area on Friday with the help of Iraqi forces who are fighting to retake the city from the Islamic State jihadist group, officials said.
Iraqi forces launched an operation to recapture Falluja, an IS stronghold located just 50 kilometres west of Baghdad, at the start of this week.
But few of the estimated tens of thousands of civilians inside the city have managed to escape.
"Our forces evacuated 460 people... most of them women and children," said police Lieutenant General Raed Shakir Jawdat.
"Hundreds of families from the people of Fallujah have been able to leave," said Raja Barakat, a member of the security committee for Anbar province, where Falluja is located.
Umm Omar, who was accompanied by more than 10 members of her family, said they were trapped in the Al-Sijr area on the northern outskirts of the city.
IS prevented them from leaving, and "gave us food that only animals would eat," Umm Omar said.
An official from the Norwegian Refugee Council said that most families who have managed to escape "were displaced from areas around Falluja... and a few were displaced from inside the city."
The NRC said in a statement earlier in the day that of the 150 families it knew of that had escaped, all but one were from the outskirts of the city.
"The situation inside Falluja is getting critical by the day," said Nasr Muflahi, NRC's Iraq director.
"We are now hearing reports of contaminated water being used for drinking, while entire neighbourhoods are being displaced within the battle zone with no safe way out," Muflahi said.
Anti-government fighters seized Falluja in early 2014, and the city later became an IS stronghold.
The jihadists overran large areas north and west of Baghdad in 2014, but have been on the defensive for months and have lost significant ground to Iraqi forces.