For youtube, you can put the video in theater mode, which makes the video the full width of your window, and pushes recommendations down below it. With this I only ever see recommendations at the end of the video.
As a general solution for us techies, you can have user defined style sheets that selectively override the site's CSS, either using a plugin like Stylus, or Firefox's built-in userContent.css. Inspect the website, find the id name (or class if it is unique enough) for the content you want to go away and put the following in your user CSS.
#<id> {
display: hidden;
}
I have so many of these. There is some upkeep with redesign, and for some sites with high churn I've given up, but in general it makes the web much more tolerable.