militants using civilians as cover, and were holding half a dozen city neighborhoods seized in the last 10 days.
The elite Counter Terrorism Service troops broke through Islamic State defense lines to enter the city early last week and have since been embroiled in a brutal, close-quarter combat with waves of suicide bombers and snipers.
The special forces are the spearhead of a wider coalition of 100,000 fighters seeking to crush a few thousand Islamic State jihadists who have ruled Mosul, the biggest city of their cross-border "caliphate" in Iraq and Syria, for the last two years.
The campaign, nearly four weeks old, is the most complex military operation in Iraq in the 13 years of turmoil since the U.S. invasion that toppled Saddam Hussein.
Security forces and army infantry divisions, backed by a U.S.-led air force, are preparing to move on southern and northern districts of Mosul in coming days, to step up pressure on the militants.
Kurdish peshmerga and Shi'ite paramilitary forces are holding territory to the northeast and to the west.
On the eastern front, special forces pushed into the Qadisiya al-Thaniya district, on the northern edge of the small pocket of neighborhoods they control so far, Sabah al-Numani, spokesman for the Counter Terrorism Service, told Reuters.
"We have encountered heavy resistance from the enemy," he said, with what he called "obstructive patrols" of militant forces trying to hold up the advance.
"We are facing the most difficult form of urban warfare, fighting with the presence of civilians, but our forces are trained for this sort of combat."
Military officers have told Reuters that the fighting is some of the most lethal they have seen, with small groups of militants using a vast network of tunnels and narrow streets to launch an apparently endless sequence of attacks against troops.
A Reuters correspondent in Kokjali, on the eastern edge of the city, saw U.S. Apache helicopters overhead. Explosions, either from air strikes or suicide car bombs which the jihadists have deployed in the hundreds since the campaign started on Oct. 17, could be heard against a backdrop of artillery fire.
As smoke rose above the city, hundreds of civilians were on the streets of Kokjali, some of them local residents but others fleeing the fighting in Mosul itself.
The International Organization for Migration says nearly 48,000 people have been displaced by the fighting, still a relatively low figure compared to a United Nations warning before the campaign of a possible exodus of up to 800,000.
Numani said the army had told civilians to stay indoors for their safety, adding that the counter terrorism unit aimed to hand over neighborhoods which it had secured to other forces. In other cities retaken from Islamic State, local police forces have moved in after the special forces have cleared territory.
KILLINGS AND CHEMICAL WEAPONS
Islamic State's two-year reign of fear in northern and western Iraq threatened the country with disintegration, and Prime Minister Haider al-Abadi says it has cost Iraq $35 billion in economic damage.
On Friday, Grand Ayatollah Ali al-Sistani praised the forces battling Islamic State, including thousands of Shi'ite fighters in the Popular Mobilisation paramilitary forces, for their sacrifices.
Without "the blood of these dear ones and their continuous steadfastness, only God would know what fate would await Iraq and others", said Seyyid Ahmed al-Safi, who delivered the Friday sermon in the holy city of Kerbala on behalf of the aged and reclusive Shi'ite religious leader.
Inside Mosul, a city which is still home to up to 1.5 million people, residents said this week that the militants had killed at least 20 people and displayed their bodies - five of them crucified - as a warning against acting as informants for Iraqi forces.
The U.N. human rights office said a total of 40 people were reportedly shot on Tuesday for "treason and collaboration" with Iraqi security forces, and a 27-year-old man was shot for using a mobile phone.
Spokeswoman Ravina Shamdasani also said the jihadists were reportedly stockpiling ammonia and sulfur in civilian areas, possibly for use as chemical weapons.
A source in the city contacted by Reuters said the militants were allowing some relatives of Islamic State supporters to evacuate and head west to Syria.
Routes out of Mosul to Islamic State's Syrian stronghold of Raqqa appear still to be open, despite efforts by the mainly Shi'ite Hashid Shaabi forces to cut them off.
The source said he had seen five families leaving Mosul. One departing person said they had permission from Islamic State and "no one will stop us on the road from Mosul to Raqqa".
In the eastern district of Karama, where fighting continued, one resident said militants were riding around on motorbikes. "We can bear the bombardment and the clashes to get rid of Daesh (Islamic State). We want to be liberated and despite all this fear we are staying in our houses," he said.
In nearby Qadisiya al-Thania, stormed by special forces on Friday, a woman said the clashes were so fierce she was too scared to go into the kitchen to cook, so she fed her family dates.
"The sound of clashes grew more distant, and then fighters reached us and raised the Iraqi flag and told us they had pushed out Daesh and liberated us," she said by phone.
"We never thought we'd be free of Daesh. We can still hear clashes and we hope they don't come back again".