There is no volume control at the bottom of youTube videos on the website or via the App.
If the speaker icon is not showing up when you adjust volume via the side switch that is not normal.
Have you tried restarting or resetting your iPad?
Restart: Press On/Off button until the Slide to Power Off slider appears, select Slide to Power Off and, after the iPad shuts down, then press the On/Off button until the Apple logo appears.
Reset: Press the Home and On/Off buttons at the same time and hold them until the Apple logo appears (about 10 seconds). Ignore the "Slide to power off"
If these don't solve your problems you may want to back up your iPad and do a Restore as new using iTunes. It sounds like your update didn't go right.