KABUL, Afghanistan — Afghanistan was plunged deeper into crisis a day after the Taliban seized the northern city of Kunduz, as the insurgents on Tuesday kept assaulting the reeling Afghan security forces and the government struggled to mount a credible response.

Not only did a promised government counteroffensive on Kunduz not make headway during heavy fighting on Tuesday, but the day ended with yet another aggressive Taliban advance, with insurgents surrounding the airport to which hundreds of Afghan forces and at least as many civilians had retreated, thinking it would be safe.

After more than a day of relative silence as the situation worsened around Kunduz, the American military showed the first signs of increased involvement in what the Pentagon called “a setback,” conducting at least two airstrikes, and reportedly more as attacks continued at the airport late Tuesday.