Iraqi forces have gradually retaken most of that territory over the past three years, and outside of Mosul the militants are largely confined to smaller towns and villages.
The advances however did slow down in November and December as the IS put up fierce resistance and used the civilian population as cover, therefore making it hard for Iraqi forces to target them.
After a lull in operations, the CTS and other forces stepped up their coordination and, with increased air and advisory support from a US -led coalition, launched a fresh push just before the New Year.
Carter said the decision to keep a residual USA military presence in Iraq would be up to the Iraqi government and Prime Minister Haider al-Abadi.
The troops made their new push into several neighborhoods in the eastern side of Mosul, locally known as the left bank of the Tigris River.
But the western part of Iraq's second city remains largely in IS hands.
But fighting in neighbourhoods in the southeast has been tougher.
"The challenge is that they [ISIS] are hiding among civilian families, that's why our advances are slow and very cautious", Lt. -Col.
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Mohammedawi said rapid response units and Iraqi army units had fought their way into the Palestine and Sumer districts in the last day, but that Islamic State fighters were firing at civilians trying to flee.
"The families, when they see Iraqi forces coming, flee from the areas controlled by Daesh towards the Iraqi forces, holding up white flags, and Daesh bomb them with mortars and Molotov cocktails, and also shoot at them".
"Whenever they (IS) withdraw from a district, they shell it at random, and it's heavy shelling", he added.
A member of Iraqi security forces in Baladiyat district, east of Mosul.
"In the 10 days since we were liberated, the bombs haven't stopped". He said Iraqi forces repelled an overnight attack, killing 37 militants, without elaborating.
The UN High Commissioner for Refugees (UNHCR) said in a recent report that the military operations in Mosul have pushed about 136,000 civilians to flee their homes in the city and its adjacent districts since the beginning of military offensive in October to reclaim the IS largest stronghold in Iraq.
"They are small drones with small munitions that they've been dropping", Sylvia said.