Currently you can only re-order your playlists if you are logged in on a desktop. Most people will be using Spotify on mobile devices so it would make a lot of sense to be able to re-order your playlists however you want on the move. Not sure why this hasn't been done yet?!
Updated on 2019-09-08
Hey @GFJallen
A similar idea has also been suggested here:
https://community.spotify.com/t5/Live-Ideas/Mobile-Rearrange-Playlists-on-Mobile/idi-p/4513814
Please vote for it, and add a comment there, so it may be considered. Any update on the matter should be posted there.
For more info on how the Ideas Board works, click here. 🙂