IS fighters seized part of a strategic hill overlooking the town of Kobane late on Saturday, a monitor said, but their progress was slowed by new strikes from the coalition of Washington and Arab allies.
A Kobane local official, Idris Nahsen, said IS fighters were just one kilometre (less than a mile) from the town and that air strikes alone were not enough to stop them.
He complained of a lack of coordination between the coalition and Kurdish fighters on the ground.
In a sign of the Kurdish defenders' determination to keep the jihadists at bay, the Syrian Observatory for Human Rights said a female suicide bomber blew herself up at an IS position east of Kobane.
"The operation caused deaths, but there is no confirmed number," Observatory director Rami Abdel Rahman said of the first reported instance of a female Kurdish fighter employing a tactic often used by IS itself.
The border town of Kobane has become a crucial battleground in the international fight against IS, which sparked further outrage this weekend with the release of a video showing the beheading of Briton Alan Henning.
The video — the latest in a series of on-camera beheadings of Western hostages — included a threat to another hostage, US aid worker Peter Kassig.
Fighting raged around Kobane as the jihadists pressed their nearly three-week siege of the town, said the Britain-based Observatory.
"IS succeeded on Saturday night in taking the southern part of the Mishtenur hill," Abdel Rahman told AFP.
He said there were seven new coalition strikes against IS positions late Saturday and that the air raids were hindering the jihadist advance.
In a statement, US Central Command said the US military carried out three air strikes in Syria on Saturday, while fighter jets, bombers and helicopters were used in six assaults against IS in Iraq on Sunday.
The sound of shelling echoed from the town — also known as Ain al-Arab — as warplanes roared overhead Sunday, an AFP reporter just across the border in Turkey said.
One mortar round hit a house on Turkish territory just a few kilometres (miles) from Kobane, wounding five people, medical sources said.
The source of the fire was unclear, but residents of two small border villages were ordered evacuated as a precaution.
Tear gas was fired to clear the border zone around the Mursitpinar crossing, the main vantage point for watching the fighting for reporters and Kurds.
Sunday's toll was not known, but the Observatory, which relies on a network of local sources, said at least 33 IS fighters and 23 of the town's Kurdish defenders were killed on Saturday.
IS began its advance on Kobane on September 16, seeking to cement its grip over a long stretch of the Syria-Turkey border.
The offensive prompted a mass exodus from the town and surrounding countryside, with some 186,000 people fleeing into Turkey.
Extremist Sunni Muslim group IS has seized large parts of Syria and Iraq, declaring a "caliphate" in June and imposing its harsh interpretation of Islamic law.
The group has been accused of carrying out widespread atrocities including mass executions, abductions, torture and forcing women into slavery.
It has also released videos of the on-camera beheadings of two US journalists, a British aid worker and on Friday of Henning, a 47-year-old British volunteer driver who went to Syria with a Muslim charity.
After first launching strikes against IS in Iraq in August, Washington has built a coalition of allies to wage an air campaign against the group.
Britain and France have joined the strikes in Iraq, while an F16 war plane carried out Belgium's first bombing raid there on Sunday.
Dutch and Australian fighter jets flew their first missions over Iraq on Sunday but did not launch any air strikes.
Five Arab nations — Bahrain, Jordan, Qatar, Saudi Arabia and the United Arab Emirates — have meanwhile taken part in the Syria raids.
French Defence Minister Jean-Yves Le Drian said Paris will "increase" the rate of air patrols over Iraq, branding IS "a terrorist army".
In Saudi Arabia, King Abdullah called for joint efforts to fight extremism "and defeat it because it has nothing to do with Islam".
Turkey's parliament last week authorised the government to join the campaign, but so far no plans for military action have been announced.
Elsewhere, the Observatory said Syrian rebels on Sunday seized a strategic hilltop in the southern province of Daraa after a two-day battle in which 30 regime forces and 29 rebels were killed.
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