Rotational dynamics of galaxies exhibits an increase beyond the Keplerian velocity which corresponds to a missing mass up to six times the dynamic mass in the observable universe. In this paper we show that the observed increase in galactic rotation velocities is a general relativistic effect resulting from the combined effect of toroidal magnetic energy density in galaxies and spacetime dragging due to the rotating compact mass in galactic center. The effect of magnetic energy density on spacetime vorticity is derived from Maxwell equations in axially symmetric spacetime where the dragging effects are shown to extend farther in the galactic disc via the toroidal field, modifying the rotational speed of the galactic matter. This is shown to lead to the diverse rotation curves of spiral galaxies, along with the Tully-Fisher relation for total galactic mass and maximum rotational velocity.