Are compiled shell scripts better for performance?

To answer the question in your title, compiled shell scripts could be better for performance — if the result of the compilation represented the result of the interpretation, without having to re-interpret the commands in the script over and over. See for instance ksh93's shcomp or zsh's zcompile.

However, shc doesn’t compile scripts in this way. It’s not really a compiler, it’s a script “encryption” tool with various protection techniques of dubious effectiveness. When you compile a script with shc, the result is a binary whose contents aren’t immediately readable; when it runs, it decrypts its contents, and runs the tool the script was intended for with the decrypted script, making the original script easy to retrieve (it’s passed in its entirety on the interpreter’s command line, with extra spacing in an attempt to make it harder to find). So the overall performance will always be worse: on top of the time taken to run the original script, there’s the time taken to set the environment up and decrypt the script.

After some googling, I found a way to compile BASH scripts to binary executables (using shc).

It's quite unfortunate that that shc contraption is still featured in google search results, even after it has been utterly debunked all these years: shc is not a compiler, and it does not prevent the source code of the script from being looked at and "stolen".

If anything, shc is even stupider than it has to be, because, after unmangling the script source, it's just passing it as an argument to bash -c, which means that it's visible in /proc/<pid>/cmdline to any user, not just the one running the script. That also runs into the Linux's length limit for a single command line argument (128k bytes). But to make things even more ridiculous, the first part of that argument is filled up with white spaces, so it doesn't appear in ps ;-)

Will it improve the performance of my script in any way?

Yes, your script may not work at all, which means that it will terminate sooner.

In general, there is no way to compile a shell script, because new source text can be introduced by several method at run time, which has therefore bypassed the compiler phase. That new source would be unable to interact with the compiled-in functions or variables.

Two methods of creating runtime source would be:

Source a side file that may have been created or modified since the original script was compiled.

Construct at runtime an arbitrary command in a string, and exec it.