A little more than a month after toppling the Western-backed government in Kabul, Afghanistan's new Taliban rulers are facing internal enemies who have adopted many of the tactics of urban warfare that marked their own successful guerrilla campaign.

A deadly attack on Kabul airport last month and a series of bomb blasts in the eastern city of Jalalabad, all claimed by the local affiliate of Islamic State, have underlined the threat to stability from violent militant groups who remain unreconciled...