  Radio free-free emission is considered to be one of the most reliable tracers
of star formation in galaxies. However, as it constitutes the faintest part of
the radio spectrum -- being roughly an order of magnitude less luminous than
radio synchrotron emission at the GHz frequencies typically targeted in radio
surveys -- the usage of free-free emission as a star formation rate tracer has
mostly remained limited to the local Universe. Here we perform a
multi-frequency radio stacking analysis using deep Karl G. Jansky Very Large
Array observations at 1.4, 3, 5, 10 and 34 GHz in the COSMOS and GOODS-North
fields to probe free-free emission in typical galaxies at the peak of cosmic
star formation. We find that $z \sim 0.5 - 3$ star-forming galaxies exhibit
radio emission at rest-frame frequencies of $\sim 65 - 90$ GHz that is $\sim
1.5 - 2\times$ fainter than would be expected from a simple combination of
free-free and synchrotron emission, as in the prototypical starburst galaxy
M82. We interpret this as a deficit in high-frequency synchrotron emission,
while the level of free-free emission is as expected from M82. We additionally
provide the first constraints on the cosmic star formation history using
free-free emission at $0.5 \lesssim z \lesssim 3$, which are in good agreement
with more established tracers at high redshift. In the future, deep
multi-frequency radio surveys will be crucial in order to accurately determine
the shape of the radio spectrum of faint star-forming galaxies, and to further
establish radio free-free emission as a tracer of high-redshift star formation.
