KABUL, Afghanistan (AP) — The Taliban vowed Tuesday to respect women's rights, forgive those who fought them and ensure Afghanistan does not become a haven for terrorists as part of a publicity blitz aimed at reassuring world powers and a fearful population.

Following a lightning offensive across Afghanistan that saw many cities fall to the insurgents without a fight, the Taliban have sought to portray themselves as more moderate than when they imposed a strict form of Islamic rule in the late 1990s. But many Afghans remain skeptical — and thousands have raced to the airport, desperate to flee...