It is the second of five bridges to be recaptured since the operation to retake the city from so-called Islamic State began in October.
The news comes a day after Iraqi forces launched a new push against the western part of Mosul.
The eastern part was recaptured in January after heavy fighting.
All five bridges were damaged in coalition air strikes but the capture of the second bridge, also known as the al-Jamhuriya bridge, will provide a foothold for government forces.
The bridges were put out of action with the aim of limiting ability of IS to resupply or reinforce their positions in the east.
Mosul is the last stronghold for IS in Iraq.
On Sunday, a senior Iraqi commander told the Associated Press that troops had been involved in the heaviest clashes since the push into western Mosul began two weeks ago.
The west includes the old city and the Great Mosque of al-Nuri, where IS leader Abu Bakr al-Baghdadi proclaimed the creation of a "caliphate" in July 2014, after the jihadist group overran large parts of northern and western Iraq.
The International Organisation for Migration says 45,000 people have fled as the fighting intensifies and aid agencies say they can barely cope with the number of people arriving at their camps.
About 750,000 people had been living in the west of the city before the new assault started.
Iraqi government forces have captured Mosul's al-Hurriya (Freedom) Bridge over the River Tigris, Iraqi military and Kurdish sources say.