According to some estimates, around 250,000 civilians are still trapped in west Mosul and can only flee once the security forces enter their neighbourhoods and clear them of Islamic State group fighters.
"The final push by Iraqi security forces to retake Mosul from (IS) has caused possibly the largest wave of displacement since the beginning of the year," the Norwegian Refugee Council (NRC) said.
Over 20,000 civilians fled their homes to a camp for displaced people in Hammam al-Alil, south of Mosul, on Thursday, NRC said, describing that figure as "a six-fold increase from the previous day".
"Everyone who is fleeing the city is in an extremely fragile state," said NRC's Iraq director, Heidi Diedrich.
"Infants and children appear malnourished and it is clear that people are suffering from hunger and trauma," she said in a statement.
NRC said the trauma hospital in Hammam al-Alil treated more trauma wounds on Friday than on any other day since it opened six weeks ago.
Iraqi forces backed by an international coalition are battling IS in the western half of Mosul and closing in on the militants in their last hideouts in Mosul's Old City.
The few hundred IS fighters left inside the city are increasingly using civilians as human shields to defend their last pockets and hoarding the city's dwindling food and medical supplies.
Around half a million people are currently displaced as a result of the offensive launched by the security forces in mid-October to retake Mosul, which was the extremists' last major bastion in Iraq.