Org-mode is sometimes breaking links while adjusting indentation in
org-fixup-indentation.
This has been observed on emacs version 27.1, with org-mode versions 9.3
(built-in) and 9.4 (melpa-stable).
It also occurs with the current emacs development version.

Here is how to recreate. I hope my mailer does not screw up the lines
below...
Enter the following text in an org buffer and proceed as described:

* Org is breaking links!
   Create a heading with `*' and random heading text.
   Create a text section below the heading.
   In the text section create a separate line starting with a link, like so:
   [[simple-link]]
   [[http://a.b.com/demo.html][link text]]
   Switch on whitespace-mode to see what is going on.
   Indent the heading deeper (i.e. demote) with several `M-<right>'
until the text starts with a TAB instead of SPCs.
   Then outdent the heading back (i.e. promote) with `M-<left>'.
   See the links getting corrupted as the two leftmost brackets `[[' are
removed, in case of links with a target part, this also gets removed.
   Note that his behavior also affects org-refile: Links can get can get
corrupted in this way while refiling.
   Note that links do not get broken as described when showing full
links with org-toggle-link-display.

Thanks for your help,
Kind Regards,
Christoph