Hello concun,
Thank you for using Apple Support Communities! I understand that when watching videos on your Mac you see a frame of the previous video. You've come to the right place for help.
To help further isolate potential causes for this behavior, first I suggest you to disable plug-ins in Safari since you mention that this behavior only occurs in Safari and not in a third party web browser:
If you need to disable or block a Safari plug-in on Mac
After following the steps, close and launch Safari and attempt to replicate the issue. If you notice the same behavior, I suggest seeing if this continues to be an issue when you restart your Mac in safe mode:
Use safe mode to isolate issues with your Mac - Apple Support
If you're able to watch videos without issue, the next step would be to rule out incompatible login items:
macOS Sierra: If you think you have incompatible login items
Testing in another user account can also help determine if this may be user or account related settings causing this issue. If you don't have another user account, here's how you can create one for testing:
How to test an issue in another user account on your Mac - Apple Support
If the behavior continues, I ask you to give us additional information about the steps you have taken and if the issue happens in other sites other than the ones you previously mentioned.
Best Regards.