Iraqi security forces launched a major push last month to recapture west Mosul, which is the most populated urban area still held by the Islamic State group, with an estimated 750,000 residents when the battle began.
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.
About 600,000 civilians remain trapped with the militants in the Mosul area, according to the International Organization for Migration.
A prominent Iraqi Sunni politician on Monday warned Washington the acceleration in a military campaign in western Mosul to drive out Islamic State jihadists was causing a sudden surge in civilian casualties that threatened to undermine the effort to crush the militants.
Staff Major General Maan al-Saadi, another CTS commander, also confirmed that the focus on Wednesday was on searching and clearing areas that had already been retaken.
"We [are] moving on the old bridge. and then we will free that area and hopefully in a few days we'll liberate the west side of Mosul", a captain from the security forces said. "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.
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Since starting the offensive in October, Iraqi forces with US-led coalition and Iranian support via Shia jihadist organisations have retaken eastern Mosul and claim to have taken about 30 per cent of the west from the militants, who are outnumbered but fiercely defending their last stronghold in Iraq.
"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.
Soldiers from the 9th Armored Division scored an important victory on Saturday night when they cut the last road out of west Mosul, said Brett McGurk, the USA envoy to the worldwide anti-IS coalition.
Displaced Iraqis flee their homes as Iraqi forces battle ISIS militants, in western Mosul, Iraq, March 6, 2017. "There has been a lot of shelling by Daesh", said Hamid Hadi, a teacher.
Losing the city would be a huge blow to Islamic State as it has served as the group's de facto capital since its leader Abu Bakr al-Baghdadi proclaimed himself head of a caliphate spanning Iraq and Syria from the Nuri Mosque in July 2014.
"The fight is in the old city now, but it's tough, the streets are narrow", said Nasser Taleb, 34, a bandage over his eye from a bullet ricochet wound he said he suffered on Aleppo Street on the edge of the old city, where intense fighting took place at the weekend.