astronomy - Experimental Data for Mass Distribution of a Galaxy

My goal here is not to discuss dark matter in general. I know there are many other observational clues that hint us towards Dark matter. My goal is simply to understand this argument here a little better.

One of the arguments for Dark matter are the observed rotational velocities of stars in the outer part of galaxies.

If we assume that most of the galaxies mass lies inside, we can neglect the outside mass and get $$ \frac{mv²}{r}=G\frac{mM(

where $M(Perkins book). This yields

$$ v \propto \frac{1}{\sqrt{r}}, $$

which isn't observed in experiments (see for example here).

Nevertheless most Galaxies are flat discs with a spherical hub in the middle and therefore maybe we can assume in the outside region for the mass density $\rho \propto \frac{1}{r}$, which seems reasonable because the mass density of a galaxy should get thinner in the outer regions.

Then we have for the mass inside radius $r$

$$ M(

and therefore using Newtonian mechanics

$$ \frac{mv²}{r}=G\frac{mM(

This is exactly what is observed in experiments.

Obviously somewhere this argument must be flawed and my best bet would be $\rho \neq \frac{C}{r}$. The standard approach seems to be to assume something of the form $\rho \propto e^{-C R} $. What experimental data shows that $\rho \neq \frac{C}{r}$ and therefore that we need Dark matter to explain other phenomena.