Two-dimensional hybrid perovskites are currently in the spotlight of condensed matter and nanotechnology research due to their intriguing optoelectronic and vibrational properties with emerging potential for light-harvesting and light-emitting applications. While it is known that these natural quantum wells host tightly bound excitons, the mobilities of these fundamental optical excitations at the heart of the optoelectronic applications are barely explored. Here, we directly monitor the diffusion of excitons through ultrafast emission microscopy from liquid helium to room temperature in hBN-encapsulated two-dimensional hybrid perovskites. We find very fast diffusion with characteristic hallmarks of free exciton propagation for all temperatures above 50 K. In the cryogenic regime, we observe nonlinear, anomalous behavior with an exceptionally rapid expansion of the exciton cloud followed by a very slow and even negative effective diffusion. We discuss our findings in view of efficient exciton-phonon coupling, highlighting two-dimensional hybrids as promising platforms for basic research and optoelectronic applications.