Kurdish forces are holding out in Kobane as a jihadist offensive enters its second month, but the Pentagon is warning US-led air strikes may not prevent the Syrian border town’s fall.

Pentagon spokesman John Kirby said the strikes had killed “several hundred” fighters with the Islamic State group and a Kurdish official inside Kobane said they had pushed the jihadists back from parts of the town.

But US officials warned that after significant advances in both Syria and neighbouring Iraq, the “tactical momentum” lay with ISIL.

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While Iraqi troops prevented the jihadists from seizing a lynchpin provincial capital west of Baghdad, a senior US envoy admitted ISIL forces had scored important advances elsewhere.

John Allen, a retired four-star general and US envoy to the coalition fighting ISIL, said it would take time to build up local forces to defeat them.

In Kobane, Kurdish official Idris Nassen said ISIL had pulled back from some areas of the town but appealed for more air strikes as well as weapons to fight the jihadists.

“The international coalition has fought ISIL more effectively during the last few days,” Nassen said by telephone.

Nassen said Kurdish forces were “flushing out” ISIL fighters from the eastern and southeastern parts of the town, calling for more military assistance.

“We need more air strikes, as well as weaponry and ammunition to fight them on the ground,” he said.

Kurdish forces have sustained heavy losses since ISIL launched its offensive on the Kurdish enclave around Kobane on September 16 but so too have the jihadists.

Ground fighting alone has killed more than 600 combatants, the opposition British-based Syrian Observatory for Human Rights said.

Between September 16 and Wednesday a total of 662 people were killed in ground fighting, said the group.

They included 20 civilians, Observatory director Rami Abdel Rahman said.

ISIL lost 374 of its militants, while 268 people have been killed fighting on the Kurdish side, he said.

US Central Command said US aircraft carried out 18 raids near Kobane over two days, hitting 16 IS-occupied buildings.

But the Pentagon spokesman warned that jihadists are pouring into the region and the town “could very well still fall”.

NATO member Turkey has stationed troops, tanks and artillery just over the border – in some cases only a few hundred metres from the fighting – but has not intervened.

It also has yet to allow US jets to mount attacks from its territory, and Turkish Prime Minister Ahmet Davutoglu said only Syrian refugees could cross into Syria to defend Kobane, rejecting US calls to open the frontier.

President Barack Obama told military chiefs from more than 20 allies this week that they are facing a “long-term campaign” – dubbed Operation Inherent Resolve on Wednesday – against ISIL.

Obama has expressed special concern for Kobane, which has become a symbolic battleground in the fight against ISIL, and about halting the ISIL advance in Iraq’s western Anbar province.

Government forces beat back an hours-long jihadist attack on the Anbar provincial capital Ramadi on Wednesday.

But Allen warned that the group has made “substantial gains” and maintained the “tactical momentum”.

The key town of Amriyat al-Fallujah, closer to the capital and one of the last parts of Anbar still holding out against the jihadists, also received reinforcements from the Iraqi army on Wednesday.

Local forces warned on Tuesday the town was in serious danger, with its police chief saying that if Amriyat falls “the battle will move to the gates of Baghdad” and the Shi’ite shrine city of Karbala.

Its fall would increase the danger to Baghdad, but ISIL fighters would still have to capture a significant stretch of government-controlled territory before reaching the capital.

“We do not believe that there is an imminent threat to the security of (Baghdad) right now,” Kirby said.