Saudi Arabia’s public prosecutor said Thursday he would seek the death penalty for five people charged in the killing of journalist Jamal Khashoggi, and the Trump administration imposed sanctions on 17 Saudi Arabian officials allegedly involved in the case.

The prosecutor also released more details about the case, saying there was no top-down order to kill the dissident journalist and distancing Crown Prince Mohammed bin Salman from it. The killing has subjected the crown prince, the country’s day-to-day leader, to intense...