Here are two main ways to get around this: using embedded codes, or using an alternative site that shows a "clean" version of the video.
When you go to a video on YouTube, go below and click on the Share tab. By default, the first tab (also labelled Share) is shown first, which has a URL. For most instances, copying and pasting this is fine, but the URL will bring you to the "official" YouTube page in all its glory...and distractions.
Instead, click Share and then click Embed:
Alternative Sites for Viewing
*Update 2/25/16: I added the directions for going to "Show More" and turning off the suggested videos at the end. Thank you Yasmine Fleming for the tip!