Through structural and optical studies of a series of two-dimensional hybrid perovskites, we show that broadband emission upon near-ultraviolet excitation is common to (001) lead-bromide perovskites. Importantly, we find that the relative intensity of the broad emission correlates with increasing out-of-plane distortion of the Pb-(μ-Br)-Pb angle in the inorganic sheets. Temperature- and power-dependent photoluminescence data obtained on a representative (001) perovskite support an intrinsic origin to the broad emission from the bulk material, where photogenerated carriers cause excited-state lattice distortions mediated through electron-lattice coupling. In contrast, most inorganic phosphors contain extrinsic emissive dopants or emissive surface sites. The design rules established here could allow us to systematically optimize white-light emission from layered hybrid perovskites by fine-tuning the bulk crystal structure.