The understanding of the transport of suprathermal ions in the presence of turbulence is important for fusion plasmas in the burning regime that will characterize reactors, and for space plasmas to understand the physics of particle acceleration. Here, three-dimensional measurements of a suprathermal ion beam in the toroidal plasma device TORPEX are presented. These measurements demonstrate, in a turbulent plasma, the existence of subdiffusive and superdiffusive transport of suprathermal ions, depending on their energy. This result stems from the unprecedented combination of uniquely resolved measurements and first-principles numerical simulations that reveal the mechanisms responsible for the nondiffusive transport. The transport regime is determined by the interaction of the suprathermal ion orbits with the turbulent plasma dynamics, and is strongly affected by the ratio of the suprathermal ion energy to the background plasma temperature.