You don't say what computer you've got or, more importantly, how old it is. When watching videos there is a lot of processing going on so, if you have a slow CPU, that might be the problem. I was having problems on my 8-year old desktop, where my newish laptop was fine. I updated the desktop and now that's fine also.
Start Windows Task Manager and have only your favourite browser showing a YouTube video running. Look at the performance graph on Task Manager. Now pause the video and look again at Task Manager and you will see the usage drop. If the usage is very high when the video is running that could be your problem, not the broadband speed.