My supernovae question I understand the collapse of a star during a supernova, as the outward pressure cannot compete with the gravitational field.

Is it conceivable, that a relaxation of the space-time curvature is what expands the space and looks like an explosion?

I have often wondered, to extrapolate, if a "release" of the curvature of space itself, or in an extreme example a massive black hole releasing all of its potential energy in an instant (In a way not yet understood) could be in fact the (or a) big bang.
No. The precise details of collapse of a supernova are not perfectly understood, in the way we understand (for example) the orbits of the planets. However the energy comes from the release of gravitational potential energy. The "relaxation of spacetime" is not significant. We don't understand supernovae perfectly. But we have a good model of them.

You should re-read Rob's answer to your previous question, not least because he got the order of magnitude of the energy release in a supernova correct. He also points out your error in thinking of the gravitational field changing "in an instant"

Concerning the big bang. No, as has been often pointed out, the big bang was not a explosion. There is no analogy between a supernova and the big bang.