The measured rotational velocities can reliably but not very accurately constrain the enclosed mass.
First, the deviation between $GM(r)/R$ and $R\partial\Phi/\partial R$ (note that this is a partial derivative at $z=0$) is typically small. Moreover, more careful modelling (decomposing the galaxy into disc and spheroidal components) usually reduces the resulting errors.
Second, even when doing so, the relation you're referring to only holds for objects on exactly circular orbits in an exactly axially symmetric gravitational potential. However, in reality even the gas does not follow exactly circular orbits, nor are galaxies exactly axially symmetric. Common deviations are spirals and bars.
If one includes all these effects into the modelling, the conclusions are not changed significantly.
When referring to the Milky Way, the rotation velocity of a circular orbit at the Solar position (known as the "Local Standard of Rest" LSR) has been measured relatively accurately to be $V_0=238\,$km$\,$s$^{-1}$ and the distance to the Galactic centre to be $R_0=8.27\,$kpc. From these numbers, the simple mass estimator gives
$$
M(<R_0) \approx R_0 V_0^2/G \approx 10^{11} \textsf{M}_\odot
$$
for the mass interior to the Solar orbit. This is larger than what one sees in terms of stars and gas, but only by about a factor $\sim2$. The measurement of rotation velocities in the outer Milky Way is notoriously difficult and the evidence for dark matter in our host galaxy is therefore based on other arguments (such as the velocities and orbits of satellite galaxies).
For external galaxies, however, one can peruse the gas velocities measured at large distances, when the observed flatness of the rotation curve (constant $V_0$ in above formulat) demands substantial amounts of matter, roughly $M\propto R$.