Syrian rebels have swept through major cities and captured territory across four provinces in a dramatic new offensive, and the Islamist leader heading the charge says their aim is nothing short of ousting President Bashar al-Assad.

After attracting little notice for years, Abu Mohammad al-Jolani and the rebels he leads shocked the world by launching the most significant challenge to Mr. al-Assad’s rule in a decade. He told The New York Times in an interview this week that he was confident the rebels would score more victories against weakened and demoralized government forces.

“Our goal is to liberate Syria from this oppressive regime,” he said in an hourlong video interview from an unknown location.

Appearing relaxed and confident, but struggling at times with a persistent cough, he was unwavering in his contention that the rebels could end the longtime Syrian leader’s brutal, authoritarian rule. “This operation broke the enemy,” he said of the rebels’ lighting offensive.

Mr. al-Jolani, 42, leads Hayat Tahrir al-Sham, an Islamist group once linked to Al Qaeda, that has controlled most of the northwestern Syrian province of Idlib for years.

He and his administration have come under criticism from inside and outside the country for using authoritarian tactics and cracking down on dissent.