Always remember that for questions involving NP, instances with a "Yes" answer are treated totally different from instances with a "No" answer. Basically, a problem is in NP if any instance with a "Yes" answer can be proven to have the answer "Yes" in polynomial time, as long as we are able to make a fantastically lucky guess to help us. But instances with a "No" answer - no idea how to prove the answer is "No".
Your Turing machine solves SAT in the slow way, without using that lucky guess. For example, in exponential time by trying out all possible combinations of inputs for the SAT problem. And you're right, based on this Turing machine, we can build another one that solves the opposite of SAT in the exact same exponential time. But that doesn't prove that either is in NP or co-NP:
To be in NP, you'd need a Turing machine that doesn't just solve SAT in exponential time. You'd need one that starts by writing a fantastically lucky hint on the tape, and then using the hint and the SAT problem, finds out that the SAT instance can be solved, and all that in polynomial time - as long as the answer is Yes. If you just reuse this turing machine to solve co-SAT, it would find solutions quick if the answer to the original SAT problem is "Yes", and therefore the answer to the co-SAT problem is "No". But that doesn't help: We need a Turing machine that can solve co-SAT problems with the answer "Yes" or equivalent, SAT problems with the answer "No", and we don't have that.