To debug performance problems, one approach is to use the built-in profiler:
M-x profiler-start RET RET
<reproduce the slowdown, ideally for a good 10s or so>
M-x profiler-report RET
Then browse the report (C-u RET is an important command there) to see where all that time is spent.
Seeing your profiler report, you'll probably want to file a bug report with the markdown-mode maintainer to see how to change the code of markdown-fontify-inline-links to try and make it faster.