Some 111,000 have sought shelter in 17 nearby camps and reception centres while many others have stayed with relatives, the ministry of displacement and migration said.
Iraqi forces backed by an international US-led coalition launched a drive to retake west Mosul on February 19 after seizing the city's eastern side the previous month.
They have since taken control of several districts including parts of the densely-populated Old City.
The Iraqi government says it can accommodate a further 100,000 displaced people in camps, but the United Nations says the numbers could rise way beyond that.
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The UN's humanitarian coordinator in Iraq, Lise Grande, said aid groups had spent months preparing for the Mosul operation.
"But the truth is that the crisis is pushing all of us to our limits," she said.
The aid operation for western Mosul is "far larger and far more complex" than in the east, she said.
"The main difference is that tens of thousands of families stayed in their homes in the east," she said. "In the west, tens of thousands are fleeing."
Iraq's second city, Mosul had an estimated population of two million before IS overran it in a lightning June 2014 assault.