You can use Improve YouTube!, an open-source browser extension that can tweak YouTube in various ways (Chrome version / Firefox version).
The videos thumbnails displayed in the sidebar on the right
Improve YouTube! can hide or collapse the right sidebar, among other YouTube UI elements.
After installing Improve YouTube!, hide or collapse the "related videos" in the right sidebar by clicking the extension's button → Appearance → Sidebar → Related videos → "Hidden" or "Collapsed". Settings screenshot:
The videos thumbnails shown at the end of playing a video
Enable: Appearance → Player → Hide endscreen to block it.
Optionally, links to other Youtube videos in the video itself, that the author has inserted.
If you're referring to YouTube annotations, enable: Appearance → Player → Hide annotations to block them.
If you're referring to YouTube cards, enable: Appearance → Player → Hide cards.
Note: I was also testing DF YouTube (Distraction Free), as was suggested in another answer to this question, but it currently (Mar. 2024) causes issues with the YouTube UI. It hasn’t been updated since Nov. 2019, which is more than 4 years ago as of this edit. Improve YouTube! is still being actively maintained, so I think it’s a better solution than DF YouTube.
With uBlock Origin (Chrome version / Firefox version), AdBlock, AdBlock Plus, or any browser extension that can use AdBlock Plus syntax filter lists, add this filter into your custom filters (called My filters in uBlock Origin):
youtube.com###related
That should block the entire sidebar including the 'Up Next' video and the list of related videos.