So at the moment the only thing we are able to do is start another thread and link to it.
That's what
should happen, in my opinion, if we had such a feature. If I were designing this feature, here's how I might have it work:
Threads can be related in a super-sub (parent-child) way.
When you reply to a thread, you have a choice to "create sub-thread". That starts a new thread, which for posting purposes is completely separate.
But the two threads are linked for forum displays and are always displayed together. When you display the main page of a forum, the New Posts list, or a search result of threads, the sub-threads of a given thread are always shown, indented, following that thread.
You could create sub-sub-threads in the same way, so the forum display would be a tree structure with many levels of indenting.
A user display option would let people suppress the display of sub-threads. A search option would let people limit searches to parent threads vs. all threads.
There would be a way to add and break the parent-child links after the fact, although maybe that would be reserved for the moderators so they could correct mistakes or detach sub-threads that outgrow their parent.