We investigate the performance of active swimmers in a strongly heterogeneous two-dimensional weakly turbulent flow. We demonstrate that there are three regimes of preferential transport for rod-like swimmers as the swimmers' intrinsic speed increases. We further reveal that the three regimes are due to the relative strengths of three different effects: the intrinsic speed of the swimmers, the reorientation ability of the shear layer at the interface of two flow regions, and the attracting Lagrangian Coherent Structures of the flow field.