Recaps of possible solutions:
1. Before going to another website, you should click on the Pause option of any video that is currently playing.
2. To fully shut down / restart the computer (this is the first step to do when trying to solve any issue):
Swipe in from the right edge of the screen, and then tap Settings.
(If you're using a mouse, point to the upper-right corner of the screen, move the mouse pointer down, and then click Settings.)
(You can also press at the same time the 3 keys Alt, Control, Delete on your keyboard, then select the shutdown/restart option.)
3. When you are in youtube website (or facebook website or any website), right-click on an empty / blank space (If you use touch, swipe up from the bottom of a touch screen or down from the top of a touch screen), then you will see all the website(s) that are currently opened, then click on the x icon at the top right of the youtube website (or any website you want) to close it. Then go to another website, then no video should be playing.
4.Instead of using Internet Explorer in the Start screen, go to the Desktop, then open Internet Explorer there on the Desktop to go to websites. It's easier to use Internet Explorer on the Desktop, then in the Start screen.
Does any of those methods work? Please click on the Reply to Expert tab below, and reply back to me, so I can help further if you need.
You said you had to go while we were chatting, so please reply back whenever you are available to continue.
Edit: It's been a few days; please let me know how it goes or if you still need help. Thank you very much.