We have acquired radio-continuum data between 70 MHz and 48 GHz for a sample of 19 southern starburst galaxies at moderate redshifts (0.067 < z < 0.227) with the aim of separating synchrotron and free-free emission components. Using a Bayesian framework, we find the radio continuum is rarely characterized well by a single power law, instead often exhibiting low-frequency turnovers below 500 MHz, steepening at mid to high frequencies, and a flattening at high frequencies where free-free emission begins to dominate over the synchrotron emission. These higher order curvature components may be attributed to free-free absorption across multiple regions of star formation with varying optical depths. The decomposed synchrotron and free-free emission components in our sample of galaxies form strong correlations with the total-infrared bolometric luminosities. Finally, we find that without accounting for free-free absorption with turnovers between 90 and 500 MHz the radio continuum at low frequency (ν < 200 MHz) could be overestimated by upwards of a factor of 12 if a simple power-law extrapolation is used from higher frequencies. The mean synchrotron spectral index of our sample is constrained to be α = -1.06, which is steeper than the canonical value of -0.8 for normal galaxies. We suggest this may be caused by an intrinsically steeper cosmic ray distribution.