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 multifrequency 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 ~ 0.5-3 star-forming galaxies exhibit radio emission at rest-frame frequencies of ~65-90 GHz that is ~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 ≲ z ≲ 3, which are in good agreement with more established tracers at high redshift. In the future, deep multifrequency 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.