When you embed a YouTube video it is good practice to restrict the related videos that appear at the end of playback to your channel only. Just imagine how your business would be affected if your competitor’s video would appear right on your own website.
Before September 25, 2018, you could completely disable related videos, but this is not the case anymore. The most you can achieve is to filter videos from other channels than from the one you are embedding the video from.
The default HTML embed code YouTube provides is something like this:
<iframe width="560" height="315" src="https://www.youtube.com/embed/dQw4w9WgXcQ" frameborder="0" allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen></iframe>
What you need to change here is the src URL. Just add ?rel=0 to the end and no related videos will be shown after the end of the video.
So instead of https://www.youtube.com/embed/dQw4w9WgXcQ which will result in this:
Enter https://www.youtube.com/embed/dQw4w9WgXcQ?rel=0 for this result:
Just fast-forward to the end of the videos to see what are the recommendations are. In the case of the second video, only videos from Rick Astley’s channel should be visible.