As fighting intensified on Tuesday after the previous day's heavy rains, civilians streamed out of western neighbourhoods recaptured by the government, cold and hungry but relieved to be free of the militants' grip.
Isis snipers were slowing the advance of Interior Ministry Rapid Response units on the Iron Bridge linking western and eastern Mosul but the elite forces were still inching forward, officers said.
Government forces also pushed into areas of western Mosul, Isis' last redoubt in the city that has been the de facto capital of their self-declared caliphate.
Federal police killed the military commander of the Old City, Abu Abdul Rahman al-Ansary, during operations to clear Bab al-Tob district, a federal police officer said.
With many Isis leaders having already retreated from Mosul, Ansary's death comes as blow to the militants as they defend their shrinking area of control street-by-street and house-by-house.
Capturing the Iron Bridge would mean Iraqi forces hold three of the five bridges in Mosul that span the Tigris, all of which have been damaged by the militants and US-led air strikes. The southernmost two have already been retaken.
"We are still moving toward the Iron Bridge. We are taking out snipers hiding in the surrounding building, we are still pushing for the Iron Bridge," Brigadier General Mahdi Abbas Abdullah of the Rapid Response force told Reuters.
Near the Mosul Museum, Iraq forces used armoured vehicles and tanks to attack snipers pinning down troops clearing areas around the bridge.
An air strike targeting one Islamic State position hit a building, engulfing nearby troops in smoke and dust.
Since starting the offensive in October, Iraqi forces with US-led coalition support have retaken eastern Mosul and about 30 per cent of the west from the militants, who are outnumbered but fiercely defending their last stronghold in Iraq.
For much of Tuesday, the troops were within 100m (330 feet) of the bridge.
"It's very key for our forces to secure the riverside and prevent Daesh militants from turning around our advancing forces," a Rapid Response spokesman said in the morning, using an Arab acronym for Islamic State.
They expected to gain control of the Iron Bridge and the nearby area by the end of the day, he said.
"Seizing the bridge will help further tighten the noose around Daesh fighters entrenched inside the old city," he said.