Optical nonlinearities are crucial for advanced photonic technologies since they allow photons to be managed by photons. Exciton-polaritons resulting from strong light-matter coupling are hybrid in nature: they combine small mass and high coherence of photons with strong nonlinearity enabled by excitons, making them ideal for ultrafast all-optical manipulations. Among the most prospective polaritonic materials are halide perovskites since they require neither cryogenic temperatures nor expensive fabrication techniques. Here we study strikingly nonlinear self-action of ultrashort polaritonic pulses propagating in planar MAPbBr$_3$ perovskite slab waveguides. Tuning input pulse energy and central frequency, we experimentally observe various scenarios of its nonlinear evolution in the spectral domain, which include peak shifts, narrowing, or splitting driven by self-phase modulation, group velocity dispersion, and self-steepening. The theoretical model provides complementary temporal traces of pulse propagation and reveals the transition from the birth of a doublet of optical solitons to the formation of a shock wave, both supported by the system. Our results represent an important step in ultrafast nonlinear on-chip polaritonics in perovskite-based systems.