The Islamic State group put up fierce resistance to defend the city it seized more than two years ago and also claimed responsibility for deadly suicide attacks further south.
The elite Counter-Terrorism Service has been spearheading the attack on the eastern front of the three-week-old offensive on Mosul, Iraq's largest military operation in years.
"Our forces are continuing to clear neighbourhoods including Al-Samah, Karkukli, Al-Malayeen and Shaqaq al-Khadra," CTS Staff Lieutenant General Abdelghani al-Assadi told AFP.
"Resistance is very heavy and they have suffered major losses," Assadi said of IS.
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Soldiers from the army's 9th armoured division also battled jihadists in the southeastern neighbourhood of Intisar, an AFP correspondent reported, as forces attempted to increase their footprint in eastern Mosul.
They first entered the streets of Mosul on Friday and were met with what one officer described as stiffer than expected resistance from IS jihadists.
Some of the first civilians to manage to escape the city proper arrived at a camp near Khazir in Kurdish-controlled territory yesterday.
Abu Sara dodged gunfire, bombs, mortar rounds and coalition strikes to flee his neighbourhood of Al-Samah, such was his desperation to leave what many civilians who escaped IS rule describe as an open-air prison.
"We walked several miles, taking with us only the clothes we were wearing and white flags we waved the entire way," said the 34-year-old, wearing a brown fake leather jacket.
The government said it had taken in 9,000 displaced people in the past two days.
The International Organization for Migration said a total of about 34,000 people had been displaced since the start of rhe offensive on October 17.