We can trust on very good authority and evidence that the Bible we have today corresponds to the original texts.
The evidence we have is the thousands of complete and partial Bible manuscripts that date very close to the actual events. Except for minor textual differences, none of which affects a major doctrine, there is remarkable similarity between ancient and modern texts.
The authority we have is God Himself. He has promised to preserve His Word, and what God promises, He is faithful to do.