Raqqa, which Daesh seized from Syrian rebels in 2014 has been a hub both for Daesh's military leaders and its bureaucrats.
US-backed Syrian Democratic Forces (SDF) have advanced into opposite sides of Daesh's so-called Syrian capital of Raqqa, the forces and a war monitor said on Saturday.
The SDF, a group of Kurdish and Arab militias supported by a US-led coalition, began to attack Raqqa on Tuesday after a months-long campaign to cut it off.
A statement by the SDF said their fighters "liberated the neighbourhood of Al-Romaniya on the western front of Raqa, after two days of continued clashes."
It said the fighting there had left at least 12 Daesh fighters dead.
The US-led coalition estimates that Raqqa, which Daesh seized from Syrian rebels in 2014 during their lightning advance in Syria and Iraq, is defended by 3,000 to 4,000.
The SDF's Kurdish elements are made up of the YPG, which Turkey considers to be the Syrian branch of the PKK, which has waged an insurgency within Turkey since 1984 that has claimed some 40,000 lives.
Raqqa has been a hub both for Daesh's military leaders and its bureaucrats and has been used to plot attacks in countries around the world.
The SDF said it had seized al-Mishlab district in the far east of Raqqa on Friday and al-Sabahia district in the west. The war monitor, the Syrian Observatory for Human Rights, said the SDF had taken Mishlab and more than half of Sabahia.
The forces are now advancing into al-Romania district in northwest Raqqa, the SDF and the Observatory said. Daesh had turned back an earlier SDF assault on a military base on the north side, the Observatory said.
Daesh still has a long sweep of territory along Syria's Euphrates valley and wide stretches of desert, despite recent losses to the SDF, the regime army and rebel groups.
Airstrikes on residential areas
Former residents in touch with relatives in the beleaguered city said at least 50 people, mostly civilians, were killed in escalating US-led strikes in the last 24 hours on residential areas inside the city.
More than 10 people were killed and dozens injured on Saturday when jets believed to belong to the US-led coalition hit the city's main Nour Street in the centre of the commercial area, they said.
The other strike that killed at least 14 people was in the western neighbourhood of Jazra, where jets targeted an internet cafe, according to Muhab Nasser, an exile from Raqqa in touch with relatives and friends.
Amaq, a news agency affiliated to Daesh, released images of what is said were white phosphorus munitions lighting the city of Raqqa that were dropped by the US-led coalition. The agency said 33 people were killed and 25 injured in a series of raids overnight.
Videos posted by Raqqa activists and personal accounts coming out of the city also cited witnesses saying the city was hit for the second consecutive night with incendiary bombs.
The US-led coalition does not deny using incendiary bombs and says casualties are unavoidable even as it tries to minimise the impact of its military campaign on civilians.
The coalition has been supplied with weapons from the US, angering Turkey which claims that the weapons could well end up in the possession of the PKK and used in Turkey.
To the west of Raqqa, the regime army and its allies have advanced into Daesh territory and on Friday reached SDF lines near the town of Tabqa, 40km (25 miles) from the city.
The Syrian regime has described the SDF's war against Daesh as "legitimate" and said its military priorities are further east, suggesting it does not plan to confront the group now.