Example of reduction such that it is not many-one reduction while it is not turing reduction
A simple example would be a reduction from e.g. $SAT$ to it's complement $\overline{SAT}$, which works as follows: given a formula $\varphi$, you can decide whether $\varphi$ is satisfiable by deciding whether $\varphi\in \overline{SAT}$, and then negating the answer. This is a Turing reduction, since you use an oracle for $\overline{SAT}$ to solve $SAT$. 

A similar example can be constructed for HALT and it's complement, if you want decidability, rather than complexity.