Magnetic fields in galaxies are believed to be the result of dynamo amplification of initially weak seed fields, reaching equipartition strength inside the interstellar medium. The small-scale dynamo (SSD) appears to be a viable mechanism to explain observations of strong magnetic fields in present-day and high-redshift galaxies, considering the extreme weakness of viable seed fields. Performing high-resolution adaptive mesh magnetohydrodynamic simulations of a small-mass, isolated cooling halo with an initial magnetic seed field strength well below equipartition, we follow the SSD amplification from supernova-induced turbulence up to saturation of the field. We find that saturation occurs when the average magnetic pressure reaches only 3 per cent of the turbulent pressure. The magnetic energy growth transitions from exponential to linear, and finally comes to halt. The saturation level increases slightly with grid resolution. These results are in good agreement with theoretical predictions for magnetic Prandtl numbers of the order ofPm ∼ 1 and turbulent Mach numbers of the order of M ∼ 10. When we suppress supernova feedback after our simulation has reached saturation, we find that turbulence decays and that the gas falls back on to a thin disc with the magnetic field in local equipartition in most of the dense gas arms. We propose a scenario in which galactic magnetic fields are amplified from weak seed fields in the early stages of the Universe to sub-equipartition fields, owing to the turbulent environment of feedback-dominated galaxies at high redshift, and are evolved further in a later stage up to equipartition, as galaxies transformed into more quiescent, large spiral discs.