Iraqi special forces battling the Islamic State reached the eastern bank of the Tigris river in Mosul on Sunday for the first time in a three-month, U.S-backed offensive to capture the city from the militants, who still control its entire western half.

The group also claimed one of two attacks at two Baghdad markets in which 20 people were killed, the latest in a spate of bombings, tactics to which Islamic State is resorting as it comes under pressure in Mosul, its last major stronghold in Iraq.

Units of Iraq's elite counter-terrorism service (CTS) have fought their way to the eastern bank of the Tigris, spokesman Sabah al-Numan said.

It was the first time Iraqi troops in the city itself have reached the river, which bisects Mosul, since the offensive to drive out Islamic State was launched in October. Iraqi forces already control the Tigris to Mosul's south.

They are not expected to push across the river without first recapturing the rest of the eastern districts, and in fact all the bridges have been taken out of service by air strikes.

But reaching the eastern bank shows the accelerated pace of the latest Iraqi advance, which has made daily gains since restarting 10 days ago.

Brett McGurk, Washington's envoy to the U.S.-led coalition backing the Iraqi offensive, said in a tweet that Islamic State's defenses in eastern Mosul were "showing signs of collapse."

The CTS has spearheaded advances inside Mosul as part of a 100,000-strong force backed by U.S. air power of Iraqi troops, Kurdish fighters and Shi'ite militias fighting the militants. After a period of stuttering advances in Mosul last month, Iraqi forces have gained momentum in a new push since around the start of the year.

CTS forces also clashed with Islamic State fighters near a historic site in eastern Mosul, a senior commander said, in a bid to drive them out of more neighborhoods.

"This morning CTS troops advanced in two directions towards the Baladiyat and Sukkar districts," Lieutenant General Abdelwahab al-Saadi said.

"During the advance, Daesh (Islamic State) tried to confront us from the historic hill," he said, referring to an elevated area near the ruins of the ancient Assyrian city of Nineveh, east of the river and inside Mosul.

Lt. Gen. Saadi said Iraqi forces and warplanes from the international coalition "dealt with" Islamic State fighters positioned on the hill, and dozens were killed.

MARKET ATTACKS

A series of air strikes sent plumes of smoke into the sky as rockets fired by Iraqi forces whizzed through the air. Sustained bursts of gunfire around midday eased as night approached.

Hundreds of civilians fled their homes in nearby Muthana, which Iraqi forces recaptured two days ago in an unprecedented nighttime raid, crossing a tributary of the Tigris via a makeshift bridge made from dirt.

Mohamed, a 35-year-old resident, said his family had locked themselves in their house for the past 10 days to avoid being forced by Islamic State to retreat deeper into the city. They walked about 3 km across the front lines on Sunday despite mortar fire from the retreating militants.

"There is no bridge, the bridge is destroyed. There is a dirt thing below the bridge. We transported our belongings and our women and our families," said Mohamed.

Life in eastern districts recaptured from Islamic State in recent weeks has regained a semblance of normalcy. Traffic wound through the streets just a few kilometers from heavy clashes, and reopened grocery stores again dot block corners.

Government-supplied electricity has cut off many in areas, but residents have begun using small generators. In some districts, they are cleaning the roads and rebuilding brick walls.

Yet in the southeastern district of Wahda, which Iraqi forces recaptured a day earlier, there was heavy damage to a hospital complex which sustained a coalition air strike and a fierce Islamic State counter-attack a month ago.

Much of the facility is burned out, some parts mined with explosives left by the militants. Blood stained the wall of one room which soldiers said Islamic State had used for executions.

In Baghdad, a suicide attacker killed 13 people when he drove an explosives-rigged car into vegetable market in the mainly Shi'ite Muslim eastern Jamila district, and detonated it, police said. Islamic State claimed the attack in an online statement, saying it had targeted a "gathering of Shi'ites."

A few hours later, a suicide bomber wearing an explosive vest blew himself up at a market in another mostly Shi'ite district, Baladiyat, killing seven, according to police and medical sources. There was no immediate claim of responsibility for the attack.

More than 80 people have been killed in just over a week in attacks in Baghdad and other Iraqi cities.

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Iraq's Kurdish regional government said in a statement on Sunday that Kurdish and coalition forces killed an Islamic State figure in a joint operation near the city of Kirkuk on Jan. 5.

The operation took place in Hawija, it said. Islamic State have a smaller presence in the area. (Additional reporting by Isabel Coles; Writing by John Davison; Editing by Raissa Kasolowsky and Alison Williams)