YouTube recently (well, more than a year ago now) changed the look of its website, bringing it into the "new age" of touchscreen-looking and impractical web design with its polymer interface. The new (er, newer) interface used to be missing several features provided by the older YouTube - you couldn't re-arrange videos in playlists, for example - though now it's much more competent for everyday use. If you still want to keep using the older interface, though, don't worry.
If you append a YouTube URL with "?disable_polymer=1" ("?disable_polymer=true" also works) it loads the original pre-polymer design - unless the URL already has a "?" in it, in which case you need to append "&disable_polymer=1". It's that easy, fortunately. As much as some people hate the polymer design, you have to laud the YouTube staff for not completely borking the option to get rid of it.
Appending a string at the end of a URL is cumbersome. Although the latter instruction is the most secure at the end of the day (businesses and government entities, take note!) if you're an end user you can get away with installing a third-party browser add-on to do this work for you. u/AyrA_ch on Reddit made a userscript to do this very task.