Islamic State (Isis) fighters are closing in on the centre of besieged Kobani, where the Kurdish militia have sworn that they will fight to the death, and hundreds of desperate civilians are trapped in streets rank with the smell of rotting bodies.

The extremist group is trying to cut off the city’s border crossing into Turkey, its last link to the outside world, and penetrate the western enclave where the Kurdish People’s Protection fighters (YPG) are most firmly entrenched. Those units stopped at least five suicide car bombs sent to blast through their last layers of defence in the past two days, activists and politicians inside the city said. But Isis is throwing fighters and ammunition at the exposed road to the border, and if that falls it would be a devastating blow to the Kurdish units.

“If they cut off the border, then everyone inside is going to die,” said activist and journalist Mustafa Abdi, who lived in Kobani until a week ago and edits the website kobanikurd.com.

“Isis can’t walk or shoot their way into the YPG strongholds, but if they can get their car bombs in it will do terrible damage. So far they have stopped them all with rocket-propelled grenades.”

The staunch defence has stemmed, but not stopped, Isis’s brutal advance through the city. On Friday the group took control of the government section of the city, including the main police station and town hall.

The UN warned of a massacre if the city falls, because even after a huge exodus of more than 200,000 refugees to Turkey there are still hundreds of civilians trapped inside. Two of them begged for a rescue mission in phone calls yesterday, as the battles raged through a powerful sandstorm that shrouded the city from journalists and anxious refugees who have been watching the fighting from the safety of Turkish soil, just a few hundred feet away.

“There is a terrible smell from bodies in the street. At first I didn’t know what it was,” said Welat Shaheen, a farmer who stayed in his home at the edge of the city when the rest of the family fled. “There are bombs and fighting all around, so no one really goes out.”

The 31-year-old is surviving on bulgar wheat and other dried food, eking out a tank of water stored up before the siege began. “I can’t wash myself, or wash dishes; it’s just for cooking and drinking. Please can someone come and get us out. If my water runs out, I will die.”

Another civilian, disabled engineer Berkal Karan, said he was eating only one meal a day to stretch out supplies. “I would like to leave, but everyone here now is trapped. When I hear voices I don’t know if they are Isis or YPG, so I am afraid to go out of my house.”

If Isis can take Kobani, it would give it full control of a long stretch of the Turkish border and a direct link between its stronghold of Raqqa in the east and positions in Aleppo province. It would also be a propaganda victory after its promises to hold prayers for the Muslim festival of Eid last week were derailed by the surprising strength of Kurdish resistance fighters.

Underlining the ideological gulf between Isis and its opponents, Kurdish fighters in the city are commanded by a woman, Heval (Comrade) Narin. Her forces have defied expectations by holding off Isis for more than 20 days, despite a paltry arsenal of light weapons that are no real match for their enemy’s huge array of heavy weapons, much of it raided from Iraqi army bases that the extremist group captured this summer.

The reputation of Kurdish forces may be bolstered by an efficient propaganda machine, but there seems little doubt that the men and women currently fighting in Kobani do it with the full knowledge that they are staring a brutal death in the face. A female fighter who was recently brought back to Turkey for burial had not just been decapitated, but had also had her breasts cut off, said Mehdi Aslan, head of a self-defence unit on the Turkish side of the border formed to stop the Isis fighters or supplies slipping into Syria.

Executions and mutilations appear to have only strengthened the resolve of fighters such as Azadin, a father of five who was ordered to leave for Turkey around a week ago because he has a family to support. He refused, saying that he would shoot himself rather than leave, relatives said. “Please don’t call, I’m fighting,” said a terse message on his voicemail when the Observer tried to contact him yesterday.

Despite their resolve, the group is now running low on ammunition and other supplies, mostly because the Turkish border has been tightly sealed for anyone wanting to travel into Syria. US air strikes, cheered by refugees watching the fight on the other side of the border, have helped to delay the Isis advance by taking out some of their largest guns.

“We are getting stronger,” said Anwar Muslim, a lawyer and head of the city council, who stayed on in Kobani after most of the officials left. “What we wanted from the beginning was to get rid of the heavy weapons so we can fight honestly. They tried everything to get inside [Kurdish-controlled areas], but for now they are still outside.”

However, even American officials have admitted that the air strikes alone are unlikely to save the city, with Isis being too well ensconced among its buildings to be bombed out. So the Kurds are desperately calling for further intervention, warning the world of a catastrophe that some fear it might already be too late to stop.

“For now, we consider Kobani lost, but we keep working and working,” said Abdi, the refugee activist. “It has been in the spotlight. People are watching it burning in front of their eyes and doing nothing. That’s still better than Kobani falling and dying with no one knowing about it.”