US-backed Kurdish forces have fought against the Syrian army and militias in the north-east of the country for a fifth consecutive day, threatening to open a new front in the five-year war.

The clashes left at least two dozen civilians dead and caused people to flee to the surrounding countryside, days after the Syrian army began using planes and artillery to attack Kurdish areas in Hasaka.

Dilan Farhad, 25, a student from the city, told the Guardian: “The Syrian army started bombing our neighbourhood and we had to flee for our lives. The cries of a woman and her child are still ringing in my ears. They were wounded by the shelling.”

As regional dynamics and alliances shift, the fighting in this previously obscure city could undermine a the Obama administration’s campaign – of which the Kurds are a key component - to degrade and destroy Islamic State.

It also risks bringing the US into a more direct confrontation with Syria in defence of its Kurdish allies and American personnel based alongside them.

The US scrambled planes over Hasaka last week after Syrian jets attacked Kurdish positions near the city, warning that it would not tolerate any threat to the 300-plus special forces stationed in north-east Syria. Speaking to CNN, Lt Gen Stephen Townsend, the US commander for Syria and Iraq, said: “We will defend ourselves if we feel threatened.”

An official within the Democratic Union party (PYD), one of the dominant Kurdish parties in Syria, said the warning did not go far enough. “The Pentagon took a stand earlier on, but we need more,” Gharib Hasso told the Guardian. “The silence of the international coalition is not good. The war is expanding.”

For the past five years, the Kurds have carved out a region in the north amid the chaos in Syria, battling groups including Isis with their YPG militia while avoiding direct confrontation with Syrian government forces.

The Kurds’ resistance against Isis won favour with the US, which first intervened on the side of the YPG with airstrikes in September 2014, when the militants looked poised to overrun the border town of Kobani.

Since then, cooperation between the two sides has deepened, with the US supporting the Kurds as part of the broader Syrian Democratic Forces. Backed by the coalition, the SDF recently recaptured the city of Manbij from Isis. On Sunday, Kurds in Turkey said a suspected suicide bombing that killed at least 50 people at a wedding in Gaziantep should be considered “a revenge attack” for the territorial gains of Kurdish forces there.

For Turkey, the growing influence of the Kurds and existence of a Kurdish entity on its southern border is unacceptable. Ankara views the YPG as an extension of the Kurdistan Workers’ party (PKK), which has fought a three-decade insurgency against Turkey.

Turkey has repeatedly complained to the US about its support for the YPG. On Sunday, the foreign minister, Mevlüt Çavuşoğlu, said the government would “wipe out” the group. The previous day, the Turkish prime minister, Binali Yıldırım, said Turkey would play a more active role in Syria. “It means not allowing Syria to be divided along ethnic lines,” he said.

The fighting in Hasaka comes as relations thaw between Turkey and Russia, which are on opposite sides of the conflict, with Moscow a key ally of Damascus. Russia has previously supported the Kurds as a means of putting pressure on Ankara, which backs the rebels fighting the Syrian president, Bashar al-Assad. Now that the two countries are on speaking terms, this could change.

Abdula Salam Ahmad, a PYD official, said: “This was a result of a consensus between Russia, Iran, Syria and Turkey to foil the gains the Kurds have made in Syria. Turkey is not happy with the recent victory in Manbij ... Turkey wants to stop the Kurds.”

In the early days of the uprising, Assad withdrew most of his forces from Kurdish areas to focus on fighting rebels, leaving a small contingent of soldiers to man government bases in Hasaka and Qamishlo.

Tensions between the two forces have occasionally escalated into violence, most recently in April, when there were skirmishes in Qamishlo until a ceasefire was brokered.

A YPG commander in Hasaka blamed Syrian soldiers for the latest flare-up. “The Syrian soldiers in Hasaka constantly harass residents, but for the past week, the harassment intensified,” Lawend Rojava told the Guardian. “We warned the soldiers, but it had no impact and that was when the fighting broke out.”

The Kurds say they have taken over most of the city and are intent on driving the Syrian army out of Hasaka.

The Iranian paramilitary Basij militia and Hezbollah are aiding Syrian forces in the two neighbourhoods of Hasaka that they still control. “The Iranians are overseeing the operation, but Hezbollah are fighting side by side with the Syrian soldiers,” said Rojava.

Footage broadcast by a YPG-affiliated channel on Sunday showed Kurdish forces engaged in street battles with Syrian soldiers in Hasaka.

One of the fighters, speaking as he and three others dashed across a road strewn with bullet casings to avoid Syrian army sniper fire, said: “Neither with planes nor with tanks can they break us. We will continue our struggle.”

Another fighter carrying a machine gun told the Ronahi TV channel: “They must walk over our bodies if they want to conquer Hasaka. We will do to the Syrian army what we did to Daesh [an Arabic term for Isis].”

Dana Razgayi contributed to this report