The intrinsic thermal (free-free) and non-thermal (synchrotron) emission components that comprise the radio continuum of galaxies represent unique, dust-free measures of star formation rates (SFR). Such high SFR galaxies will dominate the deepest current and future radio surveys. We disentangle the thermal and non-thermal emission components of the radio continuum of six ultraluminous infrared galaxies (LFIR > 1012.5 L⊙) at redshifts of 0.2 ≤ z ≤ 0.5 and 22 IR selected galaxies. Radio data over a wide frequency range (0.8 < ν < 10 GHz) are fitted with a star-forming galaxy model comprising of thermal and non-thermal components. The luminosities of both radio continuum components are strongly correlated to the 60 μm luminosity across many orders of magnitude (consistent with the far-IR to radio correlation). We demonstrate that the spectral index of the radio continuum spectral energy distribution is a useful proxy for the thermal fraction. We also find that there is an increase in mean and scatter of the thermal fraction with FIR to radio luminosity ratio which could be influenced by different time-scales of the thermal and non-thermal emission mechanisms.