Beyond that, residents and officials in Baharak, the latest district center to fall, said that more than 1,000 members of the Afghan security forces who gathered on Friday morning had retreated as the Taliban attacked and seized the area. At 5:45 p.m., the American military conducted airstrikes on both the district of Baharak and its neighboring district, an American military spokeswoman said.

Residents of Baharak interviewed by phone said that the Taliban’s shadow governor in Badakhshan had been killed in one of the strikes, but that could not be confirmed and such reports often turn out to be wrong.

In two other northern provinces, Takhar and Baghlan, the insurgents also made substantial gains in recent days. Those losses raise the question of whether the Afghan security forces, already struggling to respond to the Taliban’s capture of Kunduz at the beginning of the week, would be able to contain emboldened insurgent forces across northeast Afghanistan.

As recently as last week, the Taliban and the security forces appeared to be in a stalemate at the conclusion of a bloody fighting season that had led to a heavy death toll for Afghan soldiers and police officers.