Yes. to put it in the easiest possible way, consider this; If movement is relative, it is equally right to claim that the earth revolves around the sun, than that it is the sun that is going around the earth.
Let me attempt a more detailed explanation, I'm far from expert in the field, there are other members in here with the appropriate qualification. That said, GR is one of the most successful theories of our times, it has been able to prove its strength against a large series of experimental work.
Now, one of the pillars of the theory is that there are no "absolute inertial frames", and what this means is that the laws of physics are relative to the acceleration of every particular frame in relation to the inertial frame of our choice, and so, the idea about the earth "really" going around the sun, presented as an "objective reality" of some sort, was based on the idea that there was something immovable (the absolute inertial frame), like the earth was before Copernicus et al. Now the very concept is seen as obsolete, as movement can only be defined in relation to other bodies.