Only 800 people have been able to flee Fallujah since Iraqi forces launched a major offensive to retake the city, the United Nations said in a statement released Thursday.
Lise Grande, the UN's humanitarian coordinator for Iraq, said that those who managed to leave the city occupied by the Islamic State group reported dire living conditions inside.
"We are receiving distressing reports of civilians trapped inside Fallujah who are desperate to escape to safety, but can't," the statement quoted her as saying.
The UN said that only 800 people had been able to flee Fallujah since May 22, "mostly from outlying areas".
"Some families report having to walk for hours under harrowing conditions to reach safety. People trapped in the city centre are thought to be most at risk -- unable to flee," the UN said.
Grande said that those who managed to flee told of a dire situation inside the city, which lies only 50 kilometres (30 miles) west of the capital Baghdad.
"Food supplies are limited and tightly controlled. Medicines are exhausted and many families have no choice but to rely on dirty and unsafe water sources," she said.
The UN and other humanitarian agencies have been unable to deliver much of the available assistance due to the lack of access since the operation was launched on May 22-23.
Humanitarian corridors discussed with the Iraqi authorities have largely failed to materialise so far.
Jihadist fighters holed up in the Fallujah city centre have been imposing a curfew and forbidding residents to leave their homes, apparently using them as human cover.
Residents contacted inside Fallujah have also said that the amount of bombs and booby traps laid by IS in and around the city would make any flight very perilous.
The UN's refugee agency also said on the first day of the operation that supply routes were effectively cut off by the tens of thousands of Iraqi forces surrounding the city, thus also preventing civilians from leaving.
Various rights and other groups had warned the Iraqi government against resorting to starvation tactics to defeat IS in Fallujah, where the UN estimates around 50,000 civilians remain.