I find many websites leave the videos they add on their posts to end with choices to watch others that may cause the visitor to leave the website.
There is an option when you go to embed a YouTube video onto your post where you uncheck this and it's Show suggested videos when the video finishes.
Once you uncheck the box it automatically sets it back to the beginning. When I first started adding videos to my website, I had my first two end with the option to watch others. Once I found out you can stop this, I had to remove the videos, and start over.
I wrote a post about it back in July/17 and will leave the link here:
YouTube made a couple of changes since on how to do this but when go to make the adjustment you will see it's easy to find.
Eric has a great tutorial on how to embed a YouTube video to your website post, it's fast and easier than you think. He even shows how to center it on your post among other things. It's only 4 minutes long :D
I added a video to my recent post and it didn't have the option to uncheck a box, so I checked with 'preview' in edit. I fast forwarded it and it already had the auto-start which was nice.
Using relevant videos is great for your bounce rate, engagement and adds value to your post!