For those who constantly use video playback on the Mac, this proves to be a big problem. The alternatives can be either to switch to Chrome, but this hypothesis is discarded, or else to find a solution to the problem.
Fortunately, the solution to reproduce smooth videos on Safari it exists and easy to achieve. Follow this simple guide.
Phase 1: Click on the Safari menu and select Restore Safari.
Step 2: Make sure all options are selected in the Reset Safari window.
Step 3: Click Restore.
Safari will close and reopen. All your cookies will be deleted, and all other data associated with Safari will be deleted and restored; basically a new beginning for Safari.
If you don't want to delete all cookies, that's understandable. You can go to Safari> Preferences> Privacy> Details> select youtube.com and click on the Remove button. This method will only remove cookies and YouTube data from your Mac instead of deleting everything.
Now try to play the video in format Flash on Safari that so far has given problems, and you will notice a fluid reproduction. It worked for me, did it work for you too?