ANTAKYA, Turkey — Islamic State fighters waged street-by-street battles against militiamen in embattled Kobane on Thursday after pushing into the heart of the Syrian border town despite intense U.S. airstrikes.

The militants now control up to a third of Kobane and have fanned out toward the nearby Turkish frontier in apparent ­attempts to cut off escape routes for the town’s Syrian Kurdish defenders, according to Kurdish activists and a human rights group.

The struggle for Kobane has encapsulated the multiple issues at play within the international coalition aligned against the ­Islamic State.

Turkey’s powerful military remains within sight of the battles but has refused to cross into Syria to join the fight as Turkish leaders press demands, including a ­buffer zone in Syria along the border.

The U.S.-led coalition has stepped up attacks in a bid to push back the Islamic State. Yet air attacks alone are unlikely to halt the group’s ground advances, and U.S. officials have stressed that the air campaign must stay focused on broader strategies, such as targeting the Islamic State’s oil smuggling and supply networks.

The U.S. Central Command reported 14 more airstrikes around Kobane on Thursday, bringing the total to more than 30 this week.

Kurdish activists in contact with fighters inside Kobane said battles were taking place around the headquarters of the militia forces defending the town. Other Islamic State fighters, meanwhile, pushed north toward the Turkish border in an apparent attempt to cut off the Kurds’ escape route and encircle the town, said Mustafa Abdi, a Kurdish activist on the border.

Such a pincer movement raises fears that trapped Kurds could face the kind of retributive killings and abuses that were reported after the Islamic State captured other areas in Iraq and Syria.

The Islamic State now controls more than a third of Kobane, the Syrian Observatory for Human Rights said Thursday. Taking the town would give the militant group a greater grip over the Turkish-Syrian border and could escalate an already severe humanitarian crisis, given the flight of refugees in recent weeks.

In Turkey, the U.S. envoy for the coalition, retired Gen. John Allen, was expected to hold talks in the capital, Ankara. The Obama administration appears increasingly frustrated by Turkey’s refusal to play a stronger role in confronting the Islamic State — either by sending in its own military or boosting support for Kurdish Syrian fighters.

Turkish Foreign Minister Mevlut Cavusoglu countered that a unilateral ground operation by Turkish troops would not be enough to halt the militants’ advance.

Speaking to journalists in Ankara, he reiterated that Turkey will not join the fight unless the coalition expands its goals to include the ouster of Syrian President Bashar al-Assad.

Map: U.S.-led airstrikes on the Islamic State and the al-Qaeda Khorasan Group in Syria.

“As long as Assad stays in power, bloodshed and massacres will continue,” Cavusoglu said. “The Assad regime is the cause of instability, and, therefore, a political change is necessary.”

In Washington, State Department spokeswoman Jen Psaki said Assad’s ouster was not on the coalition’s agenda.

“That is not the focus of our international coalition and the focus of our efforts,” she said.

Among Turkey’s other demands is the creation of an internationally protected buffer zone in Syrian territory.

On Wednesday, Secretary of State John F. Kerry and several European allies said a buffer zone was worth considering. But it also could risk inflaming tensions inside Syria and embroiling the coalition in a fight with Assad’s regime, whose warplanes routinely bomb rebel-held areas of northern Syria.

Even while the airstrikes escalate around Kobane, Kerry stressed that Washington and its allies must not stray from the wider goals of trying to destroy the Islamic State’s lifeblood, such as captured oil sites and supply lines.

“As horrific as it is to watch in real time what’s happening in Kobane, it’s also important to remember that you have to step back and understand the strategic objective,” he said Wednesday in Washington.

Kurds in Turkey and Europe have staged protests demanding that Turkey intervene in Kobane. The death toll rose to 29 in three days of clashes in southeast Turkey between security forces and Kurds protesting Ankara’s inaction. In Germany, which has a significant Kurdish population, protesters waged street battles with Islamist hard-liners in Hamburg.

Murphy reported from Washington. Carol Morello and Karen DeYoung in Washington and Daniela Deane in Rome contributed to this report.