It doesn't quite work like that. For one thing, stars' orbital speeds, while reasonably fast by human standards (often hundreds of km/s), are (in most cases) incredibly slow by relativistic standards - in other words, they are miniscule fractions of the speed of light. So the difference between the "relativistic mass" (or energy, as I would call it) and the rest mass is entirely negligible for all except perhaps a few stars in any given galaxy. Certainly there's no way it could account for the missing mass attributed to dark matter.
Besides, it's not the case that the stars' orbital speeds steadily increase over time in response to increased gravity. Instead, there is going to be some equilibrium at which the effects of the increased orbital speed balance out the effects of the increased gravity. The stars will quickly reach that equilibrium during the galaxy's formation, and then, simply speaking, they will remain at that speed, so the entire galaxy exists in a steady state of orbital motion. There would be no further correlations of orbital speed with age beyond this point.
And anyway, the models that people used to determine the presence of dark matter do take this effect into account (in the sense that they have determined that it has no noticeable effect on the calculations).