Surface phonon-polaritons (SPhPs) are unique heat carriers formed by the combination of optical phonons and surface electromagnetic waves. They are mainly evanescent in the out-of-plane direction but propagate along the surface of polar dielectric membranes. In our study, we demonstrate the existence and substantial contribution of SPhPs in thin silica films through theoretical calculations. We find that these waves can propagate over long distances, reaching hundreds of micrometers, which is much longer than the mean free path of acoustic phonons. This property enables SPhPs to conduct thermal energy over a similar range as phonons when the membrane thickness is reduced below 100 nm. Our experimental results confirm that SPhP thermal conductivity becomes predominant at 800K in silicon nitride membranes over a wider temperature range. Finally, we demonstrate the size dependence of thermal conductivity due to the ballistic contribution of SPhPs.