Is the language $\{a^nb^{2^n}|n\geq 0\}$ Turing recognizable/decidable?
You are right about the fact that being able to conceive a Java program for doing the job, then there must exist a Turing machine which will also recognize that language. However, you don't need the Church-Turing thesis to confirm your statement. You need something weaker, namely the fact that both Java and TM are acceptable programming systems and hence they are isomorphic (ie. they compute the same set of partial functions) by Roger's theorem.

The main point is that Roger's theorem is a theorem and Church-Turing thesis is not proven.