A new and dangerous extremist group spawned from al Qaeda is consolidating power in northwestern Syria, while the U.S. has focused on fighting remnants of Islamic State elsewhere in the country and striking the Assad regime’s chemical-weapons facilities.

Since surfacing as the country’s most potent militant group, Hayat Tahrir al-Sham has battled Western-backed rebel groups to extend its control across Idlib province, enforcing its version of Shariah and raising funds by taxing flows of people and goods.

The...