Well... you never really prove a theory, the best you can do is find additional evidence to support it. You can disprove one though, by finding convincing evidence that directly contradicts it.
As far as String Theory goes, we still have no idea whether it has any relationship to reality, or if it is just an interesting mathematical construct. A big problem is that it is very difficult to test any of the things that String Theory implies, since its purported effects manifest at extremely high energies and very small sizes (well beyond the capabilities of current particle accelerators). So we don't have any observational evidence to back it up (yet).
OTOH, observational evidence for the Big Bang is pretty strong. Strong enough that most cosmologists believe it really happened.
If the world isn't making sense to you, you're either drinking too much or not drinking enough.