The ultrafast polarization response to incident light and ensuing exciton/carrier generation are essential to outstanding optoelectronic properties of lead halide perovskites (LHPs). A large number of mechanistic studies in the LHP field to date have focused on contributions to polarizability from organic cations and the highly polarizable inorganic lattice. For a comprehensive understanding of the ultrafast polarization response, we must additionally account for the nearly instantaneous hyperpolarizability response to the propagating light field itself. While light propagation is pivotal to optoelectronics and photonics, little is known about this in LHPs in the vicinity of the bandgap where stimulated emission, polariton condensation, superfluorescence, and photon recycling may take place. Here we develop two-dimensional optical Kerr effect (2D-OKE) spectroscopy to energetically dissect broadband light propagation and dispersive nonlinear polarization responses in LHPs. In contrast to earlier interpretations, the below-bandgap OKE responses in both hybrid CH3NH3PbBr3 and all-inorganic CsPbBr3 perovskites are found to originate from strong hyperpolarizability and highly anisotropic dispersions. In both materials, the nonlinear mixing of anisotropically propagating light fields results in convoluted oscillatory polarization dynamics. Based on a four-wave mixing model, we quantitatively derive dispersion anisotropies, reproduce 2D-OKE frequency correlations, and establish polarization-dressed light propagation in single-crystal LHPs. Moreover, our findings highlight the importance of distinguishing the often-neglected anisotropic light propagation from underlying coherent quasiparticle responses in various forms of ultrafast spectroscopy.