||We examine the role of a silicon-based amorphous insulating substrate in the thermal relaxation in thin NbN, InOx, and Au/Ni films at temperatures above 5 K. The samples studied consist of metal bridges on an amorphous insulating layer lying on or suspended above a crystalline substrate. Noise thermometry is used to measure the electron temperature Te of the films as a function of Joule power per unit area P2D. In all samples, we observe a P2D∝Tne dependence, with exponent n≃2, which is inconsistent with both electron-phonon coupling and Kapitza thermal resistance. In suspended samples, the functional dependence of P2D(Te) on the length of the amorphous insulating layer is consistent with the linear temperature dependence of the thermal conductivity, which is related to lattice excitations (diffusons) for a phonon mean free path shorter than the dominant phonon wavelength. Our findings are important for understanding the operation of devices embedded in amorphous dielectrics.