We present 15-GHz Karl G. Jansky Very Large Array observations of a complete sample of radio galaxies selected at 15.7 GHz from the Tenth Cambridge (10C) survey. 67 out of the 95 sources (71 per cent) are unresolved in the new observations and lower-frequency radio observations, placing an upper limit on their angular size of ~2 arcsec. Thus compact radio galaxies, or radio galaxies with very faint jets, are the dominant population in the 10C survey. This provides support for the suggestion in our previous work that low-luminosity () radio galaxies are core-dominated, although higher-resolution observations are required to confirm this directly. The 10C sample of compact, high-frequency selected radio galaxies is a mixture of high-excitation and low-excitation radio galaxies and displays a range of radio spectral shapes, demonstrating that they are a mixed population of objects.