It is impossible to use YouTube like an audiobook or audio source because a moment after your turn off your screen or after it is dimmed off due to timeout the playback is stopped and audio is cut-off.
To workaround this problem you can either consider or another YouTube client app:
or you can consider some other browser (namely Firefox).
Here is how:
- Navigate to youtube.com
- Find and play your video
- Hit the three-dot menu button
- Tap the Request desktop site
- When the page reloads, play the video again
- Slide down the notification center to find video controlling buttons there
Depending on your system and browser version, when you hit Power button to dim the screen, the playback will either continue or stop. If it stops, simply hit the Power button again to see locked phone screen. Firefox’s video playback controls should be there.
Replicating the same steps in mobile Chrome turned out to be ineffective.
Also note that there’s now a variety of mobile Firefoxes available in Google Play, including Beta, Nightly, Daylight and Focus. So you may get lost a bit. This answer is for original mobile Firefox and this solution was tested only in this version of Firefox.