Mosul battle: 180,000 civilians flee as Iraqi troops advance
"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'."
US officials estimate that there are about 2,000 IS fighters left in Mosul.
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