The origin and evolution of cosmic magnetic fields as well as the influence of the magnetic fields on the evolution of galaxies are unknown. Though not without challenges, the dynamo theory can explain the large-scale coherent magnetic fields that govern galaxies, but observational evidence for the theory is so far very scarce. Putting together the available data of non-interacting, non-cluster galaxies with known large-scale magnetic fields, we find a tight correlation between the integrated polarized flux density, SPI, and the rotation speed, vrot, of galaxies. This leads to an almost linear correlation between the large-scale magnetic field and vrot, assuming that the number of cosmic-ray electrons is proportional to the star formation rate, and a super-linear correlation assuming equipartition between magnetic fields and cosmic rays. This correlation cannot be attributed to an active linear -Ω dynamo, as no correlation holds with global shear or angular speed. It indicates instead a coupling between the large-scale magnetic field and the dynamical mass of the galaxies, 0.25-0.4. Hence, faster rotating and/or more massive galaxies have stronger large-scale magnetic fields. The observed correlation shows that the anisotropic turbulent magnetic field dominates in fast rotating galaxies as the turbulent magnetic field, coupled with gas, is enhanced and ordered due to the strong gas compression and/or local shear in these systems. This study supports a stationary condition for the large-scale magnetic field as long as the dynamical mass of galaxies is constant.