Why isn’t SAT in coNP?

I understand why NP=coNP if SAT is in coNP (How do I prove that SAT in coNP implies NP=coNP?).

But I’m missing why the following machine doesn’t turing recognize the complementary of SAT:

Given a turing machine M that recognizes SAT, the following turing machine recognizes coSAT:

  1. Run M on the input word w.
  2. If M accepts – reject.
  3. If M rejects – accept.

Because coSAT is the language of all unsatisfiable formulas, a formula is unsatisfiable if it doesn’t have a satisfiable interpretation, which is exactly the opposite of what M outputs.

What am I missing in here?