Turing machine that decides $A=\{0^{2^n} | n \ge 0 \}$ with an alphabet of two symbols
I'm assuming that $\_$ denotes the blank character. A strategy for the TM  you are looking for is the following:

Check if the input is a single $0$, if so accept.
Replace every other $0$ with $\_$. If, during this process, the number of encountered $0$s is odd, reject.
Pack all the $0$s together. This can be done as follows:

Reset the head to the beginning of the tape (search for a $0$ preceded by two $\_$).
Locate the last $0$ (i.e., the only one followed by two $\_$) and, in the process, remember if a $\_$ is encountered before it.
If no $\_$ is encountered you are done. Otherwise continue.
Replace the last $0$ with $\_$.
Locate the first $0$ and write write $0$ before it.
Repeat from step 3.2.


Repeat from step 1.