Iraqi forces began the push to retake western Mosul earlier this month
Around 8,000 people have fled western Mosul as Iraqi government forces attempt to retake the area from Islamic State.
The United Nations announcement came amid continued heavy fighting around the city's international airport.
The UN humanitarian aid office (OCHA) said those who fled "are often exhausted and dehydrated" by the time they reach government-controlled areas south of Mosul.
It said displacement camps are being expanded to cope with increasing numbers of people.
OCHA also said at least 75 civilians had been treated near the front lines amid "high" trauma casualty rates.
UN figures suggest 750,000 civilians are trapped in the west of the city.
On Tuesday, the Iraqi army began evacuating displaced people from the village of al Salam, south-west of the airport.
Hundreds of civilians have been heading to a gathering point carrying what few possessions they can bring with them.
There, the men are separated from the women and children, before they are taken away in buses and military pick-up trucks by the security forces.
Iraqi forces began the push to retake western Mosul on 19 February, supported by the US-led coalition.
So far they have captured the city's international airport and the military base next to it, along with some neighbourhoods on the southern edge of the city.
On Monday Iraqi troops moved further into Mosul after retaking the Jawsaq neighbourhood.
Mosul is Iraq's second-largest city and has been under IS rule for more than two years, since the summer of 2014.
It is the extremist group's last urban stronghold in Iraq.
Eastern Mosul was declared "fully liberated" by Iraq in January, after three months of fierce fighting.
(Additional reporting from IRN)