Evidence of intermodal dispersive wave generation mediated by intermodal cross-phase modulation (iXPM) between different transverse modes during supercontinuum generation in silicon nitride waveguides is presented. The formation of a higher-order soliton in one strong transverse mode leads to phase modulation of a second, weak transverse mode by iXPM. The phase modulation enables not only supercontinuum generation but also dispersive wave generation within the weak mode, that otherwise has insufficient power to facilitate dispersive wave formation. The nonlinear frequency conversion scheme presented here suggests phase-matching conditions beyond what is currently known, which can be exploited for extending the spectral bandwidth within supercontinuum generation.