UN deputy representative to Iraq Lise Grande told reporters that camps set up outside Fallujah to provide shelter are now full and conditions in the city worse than anticipated.
"The people coming out of Fallujah lost everything," she said. "They are running with absolutely nothing."
There was "widespread food deprivation" in the city, which the Islamic State group has held since January 2014, she added.
Iraqi forces launched an offensive to retake Fallujah on May 22-23, but the operation has raised alarm about the safety of tens of thousands of civilians trapped in the city.
Up to 50,000 Iraqis could still be inside, Grande said.
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The safe corridor enables civilians to move out of southwest Fallujah while Iraqi forces and tribal fighters set up a second route.
Grande said she did not know when the second safe corridor would be opened.
"More than 7,000 people have come out just in the past few days," she said.
With some eight newly built camps now full, UN agencies are working to build new facilities, along with mobile clinics, and providing the displaced with drinking water and food, Grande said.
The envoy deplored the lack of international funding for the effort, saying some $60 million is needed to help the families fleeing Fallujah "and that's 60 million that we don't have right now."
The IS group has been using civilians as human shields in the center of the city and some Iraqis have come under fire during their escape.
"This is the time to stand up in solidarity with the people terrorized by ISIL," Grande said.