I'm sure this problem has been addressed before, but I don't really know how to look for it.

I've been installing many mods, such as Better Bodies, BTB's Game Improvements etc. Until now they didn't create any problems, but around the time when I installed Tamriel Rebuilt (worth noting I used it with a preexisting savegame) I got this.

Simply put, when the game is loading, whether from a hard save or via quickload, the progress bar stops in the "Initializing data" stage for around 30 seconds, which did not happen before.

How can I fix this? Is it even possible? Is it related to savegame bloating? (I'm very new to the whole scene, so I don't know much about these affairs)

First thing, try a new character, see if you get the same delay. If so, you have a mod issue.

Although 30 seconds seem longer than usual for no progress, it is still very normal that adding mods (especially large ones like TR) will make the initializing a lot longer.

It is best to make a fresh save file often. Reusing the same file over and over isn't recommended.
Also a good idea to wait 72 hours occasionally. That removes a lot of junk info the game doesn't need to remember.