In metal halide perovskites the complex dielectric screening together with low energy of phonon modes leads to non-negligible Fr¨ohlich coupling. While this feature of perovskites has been already used to explain some of the puzzling aspects of carrier transport in these materials, the possible impact of polaronic effects on the optical response, especially excitonic properties, is much less explored. Here with the use of magneto-optical spectroscopy, we revealed the non-hydrogenic character of the excitons in metal halide perovskites, resulting from the pronounced Fr¨ohlich coupling. Our results can be well described by the polaronic-exciton picture where electron and hole interaction are no longer described by a Coulomb potential. Furthermore, we show experimental evidence that the carrier-phonon interaction leads to the enhancement of the carrier’s effective mass. Notably, our measurements reveal a pronounced temperature dependence of the carrier’s effective mass, which we attribute to a band structure renormalization induced by the population of low-energy phonon modes. This interpretation finds support in our first-principles calculations.