Supercontinua are broadband spectra that are essential to optical spectroscopy, sensing, imaging, and metrology. They are generated from ultrashort laser pulses through nonlinear frequency conversion in fibers, bulk media, and chip-integrated waveguides. For any generating platform, balancing the competing criteria of strong nonlinearity, transparency, and absence of multiphoton absorption is a key challenge. Here, we explore supercontinuum generation in integrated gallium nitride (GaN) waveguides, which combine a high Kerr nonlinearity, mid-infrared transparency, and a large bandgap that prevents two- and three-photon absorption in the technologically important telecom C-band, where compact erbium-based pump lasers exist. Using this type of laser, we demonstrate tunable dispersive waves and gap-free spectra extending to almost 4 µm in wavelength, which is relevant to functional group chemical sensing. Additionally, leveraging the material’s second-order nonlinearity, we implement on-chip f-to-2f interferometry to detect the pump laser’s carrier-envelope offset frequency, which enables precision metrology. These results demonstrate the versatility of GaN-on-sapphire as a platform for broadband nonlinear photonics.