More than 180,000 civilians have fled the west of the Iraqi city of Mosul in the past month, as government forces battle Islamic State (IS) militants.
Iraqi officials say most have taken refuge in camps and reception centres. Others have moved in with relatives.
The UN has warned an additional 320,000 civilians may flee in the coming weeks.
It is also worried about civilians trapped in Mosul's densely-populated Old City, where troops are advancing despite fierce IS resistance.
Over the weekend, police units approached the Great Mosque of al-Nuri, where IS leader Abu Bakr al-Baghdadi proclaimed the creation of a "caliphate" in July 2014 after the group seized swathes of Iraq.
Iraqi government forces launched a major offensive to recapture Mosul, the last major IS urban stronghold in the country, in October.
Supported by US-led coalition air strikes and military advisers, they have managed to retake large parts of the city and its surrounding area.
They took full control of all of Mosul's eastern side in January and have driven IS militants from several districts since starting an assault on the west on 19 February.
UN humanitarian co-ordinator Lise Grande said the humanitarian operation in western Mosul had so far been far larger and more complex than in the east.
"The main difference is that tens of thousands of families stayed in their homes in the east - in the west, tens of thousands are fleeing," she explained.
The government is rushing to build new camps or expand existing facilities to accommodate the displaced but Ms Grande said it was "a race against time".
Map of Mosul territory, 20 March 2017
"If the number of people leaving the city increases faster than we can construct new plots, the situation could deteriorate very quickly," she added.
Ms Grande also warned that civilians still inside the Old City were at grave risk, with the use of explosives there likely to cause extensive damage where streets were narrow and houses packed closely together.
"Families are at risk of being shot if they leave and they are at risk if they stay," she said. "It's horrible. Hundreds of thousands of civilians are trapped and they are in terrible danger."
Families have also told the UN that shop shelves are empty. Water and electricity supplies have also been cut and medicines are running out.
"The situation is very dramatic," said Ms Grande. "We fear it will get far worse in the days and weeks ahead."
Residents who have managed to flee say the militants are using civilians as human shields, hiding in houses and forcing young men to fight.
Ali, a former government worker, told Reuters news agency: "All [the militants] are doing now is defending. I hid my sons in the basement and told them: 'If you want my sons you will have to kill me'."