As is well-known, the phonon and electron thermal conductivity of a thin film generally decreases as its thickness scales down to nanoscales due to size effects, which have dramatic engineering effects, such as overheating, low reliability, and reduced lifetime of processors and other electronic components. However, given that thinner films have higher surface-to-volume ratios, the predominant surface effects in these nanomaterials enable the transport of thermal energy not only inside their volumes but also along their interfaces. In polar nanofilms, this interfacial transport is driven by surface phonon polaritons, which are electromagnetic waves generated at mid-infrared frequencies mainly by the phonon-photon coupling along their surfaces. Theory predicts that these polaritons can enhance the in-plane thermal conductivity of suspended silica films to values higher than the corresponding bulk one, as their thicknesses decrease through values smaller than 200 nm. In this work, we experimentally demonstrate this thermal conductivity enhancement. The results show that the in-plane thermal conductivity of a 20 nm thick silica film at room temperature is nearly twice its lattice vibration counterpart. Additional thermal diffusivity measurements reveal that the diffusivity of a silica film also increases as its thickness decreases, such that the ratio of thermal conductivity/thermal diffusivity (volumetric heat capacity) remains nearly independent of the film thickness. The experimental results obtained here will enable one to build on recent interesting theoretical predictions, highlight the existence of a new heat channel at the nanoscale, and provide a new avenue to engineer thermally conductive nanomaterials for efficient thermal management.