The relationship between mass and curvature of spacetime is well-motivated, but not the consequence of some fundamental fact in GR.
There are plenty of experiments confirming the curvature of spacetime by masses, like gravitational lensing, the proper prediction of the perihelion of mercury, redshift in a gravitational field...
$E = mc^2$ is one of the most famous equations in physics, but also the single most one quoted out of context. The full equation reads
$$ E^2 = \vec p^2 c^2 + m^2 c^4$$
This implies that indeed, mass is just a form of energy and the other way round that there is no fundamental distinction (in GR) between objects with energy from rest mass or energy from momentum. This can be seen by the fact that photons DO interact with the gravitational field (see gravitational lensing), wile their rest mass $m = 0$.