"The Rapid Response force and the federal police have liberated Jawsaq neighborhood and now control the western end of the fourth bridge," said Brigadier General Yayha Rasool, a spokesman for the Joint Operations Command, in comment to AFP.
"That means the bridge is under control on both sides."
An Iraqi military source who spoke to The New Arab's Iraq correspondent Baraa al-Shammari said that Iraqi forces had engaged IS militants in the al-Dawasa neighbourhood in West Mosul.
An IS commander and three other combatants from the extremist group were killed in the clashes.
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The "fourth bridge" is one of five bridges in Mosul - all of which are damaged beyond use - across the northern Iraqi city which was taken by IS in June 2014.
Setting up a "ribbon bridge" - used to transport weapon systems, troops and supplies over water when permanent bridges are not available - is a key aim of current military pushes launched on Mosul's west bank earlier this month.
Iraqi forces took control of the east bank of the city in January, with operations combatting IS in the city beginning in October 2016.
Current offensives follow on from Iraqi forces' liberation of the Mosul International Airport on Friday, during operations that facilitated current pushes into the west of the city.
(Agencies contributed to this report)