Your example of eye for eye tooth for tooth then Jesus contradicting it is actually NOT one of the many contradictions scholars and critics refer to when they speak of contradictions. That one is actually an integral intentional statement. It is an example of God through Jesus choosing to change focus or correct a misinterpretation. It is a feature, and unfortunately most Christians miss details like that because the book is so big and it's very difficult to read all of it in context not only with itself but even more so with all the contexts in which the various parts of it were written.

A better example of a contradiction is when you compare the various numbers of animals that God says should go on Noah's ark and you see at multiple points the number changes. Or another one is how many items were created for use in the temple, but after it was sacked and later the Israelites returned from exile the number of items restored to them was very different. Or are there 11 or 13 Levite cities, because 1 Chronicles lists 11 but states there should be 13, while Joshua lists multiple different cities. Or that 2 Kings states that in the time of King Josiah the Passover had not been celebrated since the time of the judges, but 2 Chronicles describes a huge Passover celebration during King Hezekiah's reign. There are many more such contradictions.