Context. Magnetic fields, which regulate stellar feedback and star formation in galaxies, are key to understanding galaxy evolution. Aims. We probe the origin of magnetic fields in late-type galaxies, measuring magnetic field strengths and exploring whether magnetic fields are only passive constituents of the interstellar medium or whether, being part of the local energy equilibrium, they are active constituents. Methods. We measure equipartition magnetic field strengths in 39 galaxies from the second data release of the LOFAR Two-metre Sky Survey (LoTSS-DR2) at 144 MHz with 6 arcsec angular resolution (0.1-0.7 kpc). For a subset of nine galaxies, we obtain atomic and molecular mass surface densities using H I and CO(2-1) data from the THINGS and HERACLES surveys, respectively. These data are at 13 arcsec angular resolution, which corresponds to 0.3-1.2 kpc at the distances of our galaxies. We measure kinetic energy densities using H I and CO velocity dispersions. Results. We find a mean magnetic field strength of 3.6-12.5 μG with a mean of 7.9 ± 2.0 μG across the full sample. The magnetic field strength has the tightest and steepest relation with the total gas surface density, with B α ΣH I+H20.309 ± 0.006. The relations with the star-formation rate surface density and molecular gas surface density have significantly flatter slopes. After accounting for the influence of cosmic-ray transport, we find an even steeper relation of B α ΣH I+H20.393 ± 0.009. Conclusions. These results suggest that the magnetic field is regulated by a B-ρ relation, which has its origin in the saturation of the small-scale dynamo. This is borne out by an agreement of kinetic and magnetic energy densities, although local deviations do exist, in particular in areas of high kinetic energy densities where the magnetic field is sub-dominant.