An estimated 50,000 to 100,000 people are believed to be trapped inside the besieged Syrian city, the UN says.
US-backed Syrian Democratic Forces (SDF), aiming to oust Daesh from its Syrian stronghold Raqqa, captured a ruined fortress on the edge of the city on Wednesday, and a US coalition official said the attack was set to accelerate.
SDF which includes Arab and Kurdish militias, on Tuesday declared the start of its offensive to seize the northern Syrian city from Daesh, which overran it in 2014.
With tens of thousands of people uprooted by the fighting, a UN official warned of a dire humanitarian situation, with shortages of food and fuel.
"We are receiving reports of air strikes in several locations in Raqqa city," UN aid official Linda Tom told Reuters by phone from Damascus.
By Wednesday, the SDF had moved into the western outskirts of Raqqa and were trying to advance into an eastern neighbourhood. Shelling and air strikes from the US-led coalition hit targets around the city's edges, according to a war monitoring group and the YPG.
The Raqqa assault overlaps with the final stages of the US-backed attack to recapture Daesh's capital in Iraq, the city of Mosul.
Brett McGurk, the American envoy to the US-led coalition against Daesh, said it was significant that the SDF now had a "foothold" in Raqqa.
Daesh are "down to their last neighbourhood in Mosul and they have already now lost part of Raqqa. The Raqqa campaign from here will only accelerate," McGurk said in Baghdad on Wednesday.
But he said the coalition and SDF were prepared for "a difficult and a long-term battle."
Turkey has been critical of the US decision to arm and fight alongside the SDF, which is dominated by the YPG, the armed wing of the PYD, which is a Syrian affiliate of the PKK, a group listed as a terrorist organisation by Turkey, the US and EU.
The Daesh terrorist group has been forced into retreat across much of Syria. Its biggest remaining foothold is in the eastern province of Deir Al Zor, which borders Iraq.
The UN's Tom said an estimated 50,000 to 100,000 people were trapped inside Raqqa, far fewer than its population before the Syrian war erupted in 2011. Many have fled to camps elsewhere in Syria.
In some areas around Raqqa, where the SDF has recently taken control, people had started returning home, said Tom, but yet more were still being uprooted and the situation was very fluid.
YPG spokesperson Nouri Mahmoud said displaced people were coming from all edges of the city after finding their own routes out.
He said when refugees arrived at SDF positions they were being given tents and supplies, but much more humanitarian support was needed to cope with the large numbers.
"Their situation is tragic, it is difficult. There isn't much support for them," Mahmoud said.
Now in its seventh year, the Syrian conflict has killed hundreds of thousands of people and driven more than 11 million people from their homes.