It would share all the same limitations as any other forking server, plus extra overhead for any spawned child processes. Basically the same as using CGI scripts, and probably not much worse any Apached hosted PHP based website, tbh. Probably wouldn't be worth implementing HTTP/2 since it would increase the complexity with no real benefit.
Performance is kind of a meaningless question in this case, since the "bash" script makes calls to external binaries. You could write a webserver binary in C, then invoke it with bash and call it a shell script.
This is a good question. I've always kind of wondered what bash's performance characteristics were compared to other programming languages. Bash certainly feels fast.