What is wrong with my 2 bit adder?
Perhaps you were confused about the switches you set up. I'm not sure. But I think you actually got everything right. Here's a re-drawing that mirrors your own example:

I've also added a test process to validate the results, as you can see on the right side there.
So I believe you did everything correctly, except that you got mixed up on which switch belonged to which bit on the input side.