BERLIN—Turkish President Recep Tayyip Erdogan wants a German court to stop one of Germany’s most powerful media executives from continuing to defend a comedian who caused a diplomatic spat with a crude poem about the Turkish leader.

A request for a court order, or preliminary injunction, against Mathias Döpfner, chairman and chief executive of German media company Axel Springer SE, was filed in state court in Cologne, said Ralf Höcker, a lawyer representing Mr. Erdogan. He said the goal is to stop a “wave of insults” against...