I'm seeing the same thing on iOS 11.3 (GA version, not a beta) on my iPad and iPhone as well. I'm pretty sure it's a change in iOS 11.3 and the YouTube playback engine. I also see this behavior on embedded video in the Apple News app. It's not just Safari on iOS, but rather an interaction with the video playback in general on iOS 11.3.
To get around it, touch the "YouTube" icon just to the left of the AirPlay icon. This will open the video on the YouTube web site, which does let you go to full-screen mode.