Improving heat dissipation in increasingly miniature microelectronic devices is a serious challenge, as the thermal conduction in nanostructures is markedly reduced by increasingly frequent scattering of phonons on the surface. However, the surface could become an additional heat dissipation channel if phonons couple with photons forming hybrid surface quasiparticles called surface phonon-polaritons (SPhPs). Here, we experimentally demonstrate the formation of SPhPs on the surface of SiN nanomembranes and subsequent enhancement of heat conduction. Our measurements show that the in-plane thermal conductivity of membranes thinner than 50 nm doubles up as the temperature rises from 300 to 800 kelvin, while thicker membranes show a monotonic decrease. Our theoretical analysis shows that these thickness and temperature dependencies are fingerprints of SPhP contribution to heat conduction. The demonstrated thermal transport by SPhPs can be useful as a previously unidentified channel of heat dissipation in a variety of fields including microelectronics and silicon photonics.