In general relativity we express the laws of physics using 4-vectors. 4-vectors are defined to have the same magnitude in all coordinate systems. E.g. the energy momentum 4-vector is $pc = (E, \mathbf p c )$. The magnitude of this vector is mass $mc^2$, where
$$ m^2 c^4 = E^2 - \mathbf p^2c^2.$$
Mass is then an invariant quantity which does not increase with velocity. This is just as true in special relativity as in general relativity.
Unfortunately there is still confusion dating from the early days of special relativity, when momentum was still defined as a 3-vector
$$\mathbf p = m\mathbf v. $$
using the velocity 3-vector. Writing relativistic formulae using 3-vectors caused people to define relativistic mass, which increases with velocity. We now recognise that relativistic mass is simply energy, and that it is much better to always use 4-vectors, and to drop the confusing idea that mass increases. There is no confusion about saying that energy increases with velocity.
Specifically we have the velocity 4-vector
$$v= (\gamma , \gamma \mathbf v)$$
where $\gamma$ is the Lorentz factor, and we define the energy momentum 4-vector
$$(E,\mathbf p) = p = mv= (\gamma m, \gamma m\mathbf v)$$
so, taking the 3-vector part, we have
$$\mathbf p = \gamma m\mathbf v$$
which shows that it is really a mistake to define $\gamma m$ as relativistic mass, because the Lorentz factor $\gamma$ is associated with the kinematic factor $\mathbf v$, not with mass.