At least 20,000 Iraqi civilians who were besieged by jihadists on a mountain have managed to flee after US air raids on Islamic State (Isis) forces, officials have said.

Shawkat Barbahari, an official from the autonomous Kurdish region of northern Iraq, said 30,000 people had escaped to Syria and then been escorted back into Iraqi Kurdistan by Kurdish forces. A spokesman for the UN Office for the Coordination of Humanitarian Affairs (UNOCHA) in Iraq said officials had been reporting to the UN that 15,000 to 20,000 people had escaped the siege. Fears had been growing for the civilians, mostly Kurds of the Yazidi faith, trapped on Mount Sinjar in north-west Iraq in the searing summer heat with little to eat or drink.

The breakthrough coincided with US air raids on Isis fighters in the Sinjar area on Saturday. Barack Obama, who sanctioned the air strikes on Friday, has said the US is in it for the long-haul, warning "this is going to take some time".

Barbahari, who is in charge of the Fishkhabur crossing with Syria, told AFP: "The Kurdish peshmerga forces have succeeded in making 30,000 Yazidis who fled Mount Sinjar, most of them women and children, cross into Syria and return to Kurdistan. Most of them crossed yesterday and today, this operation is ongoing and we really don't know how many are still up there on the mountain."

Iraqi MP Vian Dakhil, who is from the Yazidi minority, said 20,000 to 30,000 had managed to flee the mountain they ran to a week ago when militants overran the Sinjar region and were now in Iraqi Kurdistan. "The passage isn't 100% safe," she said. "There is still a risk."

Iraq's human rights minister, Mohammed Shia al-Sudani, told Reuters that Isis militants have killed at least 500 members of Iraq's Yazidi minority during their offensive in the north. He said the Sunni militants had also buried some of their victims alive, including women and children. Some 300 women were kidnapped as slaves, he added.

"We have striking evidence obtained from Yazidis fleeing Sinjar and some who escaped death, and also crime scene images that show indisputably that the gangs of the Islamic States have executed at least 500 Yazidis after seizing Sinjar," Sudani said.

UNOCHA spokesman David Swanson stressed that the UN was not directly involved in the evacuation of the civilians from Mount Sinjar and could not confirm the numbers but stood ready to assist those crossing back into Kurdistan's western Dohuk province, where the UN has a presence.

An RAF transport aircraft made the first airdrop of British humanitarian aid on the mountain on Saturday night, joining the US in dropping clean water, filtration devices, food and tents.

The UK's international development secretary, Justine Greening, said: "The world has been shocked by the plight of the Yazidi community. They face appalling conditions, cut off on Mount Sinjar after fleeing persecution by Islamic State extremists.

"The UK has acted swiftly to get life-saving help to those affected. Last night the RAF successfully dropped lifesaving UK aid supplies, including clean water and filtration devices, on the mountain."