About one thousand civilians walked across the frontlines, which is the largest displacement since the new offensive began a week ago.
Iraqi Prime Minister Haider al-Abadi on Friday promised that Iraqi forces would do all they could to keep civilians safe.
Iraqi forces, backed by aerial support by the US-led global coalition, launched a new push last week to drive Isil from Mosul's west.
Commanders expect the battle there to be more hard, in part because tanks and armoured vehicles can not pass through the narrow alleyways that crisscross ancient districts.
Iraq's second largest city, Mosul has been under ISIS control for over two years, and the ongoing invasion of Mosul has been ongoing for several months now, with the western half of the city just now being entered by Iraqi forces, at a very slow pace.
This weekend as many as 2,500 residents of Mosul escaped from the western half of a city that has been under the yoke of so-called Islamic State (IS) for nearly three years.
Nearly a week into a major push on the city's west bank, they were gaining significant ground, taking on IS on several fronts in one of the most intense phases of the four-month-old operation to retake Mosul.
Yet the psychological impact of drone attacks can not be discounted, says Emanuele Nannini from the Italian aid agency Emergency, which helps run the hospital where Umm Mohammed and several other drone victims are being treated. A Reuters correspondent saw two militants' corpses outside a mosque in Josaq. ISIS "only has so much capacity" to mass fighters, he said, and when they gather in force, American planes, helicopters, and guided rockets strike them.
Authorities Begin Arresting Dakota Pipeline Protesters Refusing to Vacate Camp
The camp's population in recent weeks dwindled to several hundred people as the pipeline battle moved primarily into the courts. Early Wednesday, protesters burned some wooden structures on site in what they described as a leaving ceremony.
Special forces Lt. Gen. Abdul-Wahab al-Saadi said that his troops are "moving very slowly" and that IS fighters are responding with auto bombs, snipers and dozens of armed drones.
It's also thought that, in another example of drones adapted for urban warfare, IS militants are using the small aircraft to guide suicide vehicle bombers to their targets.
A Mamoun resident reached by phone said militant fighters had flooded the area in recent days while moving their families to relative safety in other districts.
"We're heading towards the center and also the Turkish consulate, which we're about 500 meters from", he said, as attack helicopters fired rockets at targets in the Jawsaq neighborhood.
A woman forced to leave Wadi Hajr district said the militants had climbed to her roof and knocked holes in the walls in order to move undetected.
Meanwhile, UN aid workers say they are extremely concerned by the humanitarian situation in remaining IS-held areas of western Mosul, where about 750,000 people are believed to be living. "They were calling to each other and saying, "Let's go".
The majority were shot and thrown into the pit, locals said.