As to my understanding, fonts which are not offered by Typst directly, have to be copied in the project folder (if the Web-App is used). But most fonts consist of quite a few styles, so the project folder gets easily polluted.
Is there any way to get a cleaner project folder, by putting the fonts in a subdirectory?
It should work. Typst discovers fonts inside the project automatically, it doesn’t make a difference whether you upload the fonts to the project root or to a subdirectory of it.
"I always create an asset directory, which typically contains a media subdirectory for all my graphics and logos, as well as a fonts subdirectory where I store all my font files. You just need to reference it correctly from your main document; in my case, for example, by using "/assets/fonts/yourfont.otf"
As far as I understand @flokl there is no need to reference fonts by their path. You just use them like pre-installed fonts?!
Ah yes, true that 
You can also reference them directly, of course.