The new heir to Saudi Arabia’s throne has launched a crackdown on dissent in recent weeks, attempting to silence activists and critical clerics as well as his deposed predecessor, according to U.S. and Saudi officials familiar with the events.

King Salman upended Saudi Arabia’s succession order last month by naming his 31-year-old son, Mohammed bin Salman, crown prince and next in line to the throne, and sidelining his nephew and heir apparent, Mohammed bin Nayef, who has deep ties to U.S. intelligence and is widely viewed...