chess online

I am not the world's greatest player, I'm learning. But why in the world would I expect that Bxg2 would prompt the suicidal move Qxb8? I would think that White's quite sensible response would be Kxg2, taking the black bishop without risk or loss. But it's late and I might be missing something obvious. So, since this is about the fourth "senseless-to-me" move that has injured my pride by lowering my score I would like to make the most of it. Would somebody please show me where I am wrong?
Thanks!