Theoretical scenarios, including the turbulent small-scale dynamo, predict that strong magnetic fields already exist in young galaxies. Based on the assumption of energy equipartition between magnetic fields and turbulence, we determine the galactic synchrotron flux as a function of redshift z. Galaxies in the early universe are different from local galaxies, in particular, the former have more intense star formation. To cover a large range of conditions, we consider two different systems: one model galaxy comparable to the Milky Way and one typical high-z starburst galaxy. We include a model of the steady-state cosmic ray spectrum and find that synchrotron emission can be detected up to cosmological redshifts with current and future radio telescopes. The turbulent dynamo theory is in agreement with the origin of the observed correlation between the far-infrared (FIR) luminosity L-FIR and the radio luminosity L-radio. Our model reproduces this correlation well at z = 0. We extrapolate the FIR-radio correlation to higher redshifts and predict a time evolution with a significant deviation from its present-day appearance already at z approximate to 2 for a gas density that increases strongly with z. In particular, we predict a decrease of the radio luminosity with redshift which is caused by the increase of cosmic ray energy losses at high z. The result is an increase of the ratio between L-FIR and L-radio. Simultaneously, we predict that the slope of the FIR-radio correlation becomes shallower with redshift. This behavior of the correlation could be observed in the near future with ultra-deep radio surveys.