[kmm]

Galactic rotational velocities are on the order of 100 km/s, which, though blazingly fast by human standards, is so slow compared to the speed of light that special relativistic effects are completely negligible.

Moreover, it's not just the absolute magnitude of the velocity that is the problem, it's also the shape of the curve. Newtonian mechanics (even with the extremely tiny relativistic corrections taken into account) predict that the rotational speed of a galaxy would decrease the further you go out from the center, and we in fact find that this speed remains more or less constant.