A pan-European effort to crack the Islamic State network behind the Paris and Brussels attacks is yielding an unsettling discovery—a web of interlocking terror cells whose dimensions authorities say they are still trying to grasp.

European authorities said they suspect that several men detained in a number of countries over the Easter weekend all had connections to perpetrators of the deadly attacks. This has prompted French and Belgian prosecutors to seek closer U.S. assistance, according to Western officials, as they try...