Iraq launched a massive operation on October 17 to retake Mosul from the jihadists, who seized the city more than two and a half years ago, and its forces have recaptured a number of neighbourhoods on the east side of the river.
The smaller but more densely populated west side of Iraq's second city remains completely under IS control.
Counter-Terrorism Service forces "are about 500 metres (yards) from the fourth bridge," spokesman Sabah al-Noman said, referring to the southernmost bridge across the Tigris in Mosul.
"This operation was precisely planned; in fact we have been working on it for almost a week," he said, adding that it "surprised the enemy."
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The CTS and the Rapid Response Division are the two elite units leading the advance against IS in Mosul.
A commander said that Rapid Response forces had recaptured the Al-Salam hospital, where troops were cut off and mauled by the jihadists early last month.
"Rapid Response forces... Are in control of Al-Salam Hospital and the medical college and Al-Shifa Hospital," Staff Lieutenant General Abdulamir Yarallah said in a statement.
IS overran large areas north and west of Baghdad in 2014, but Iraqi forces have since recaptured much of the territory they lost, and Mosul is now the country's last major city in which the jihadists hold significant ground.
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