Howdy
A couple of things. In addition to cocking the hammer with the off hand thumb, keep the pistol on target for all the shots. Raise the pistol up, get the sight picture, cock the hammer and fire. Repeat four times. Do not lower the gun until you are done shooting.
With all due respect, that video of the guy fanning his 22 is not Cowboy Action Shooting. Fanning is forbidden in CAS, his pistol has been heavily modified for fanning, and other than kids, everybody has to shoot pistols of at least 32 caliber.
Now, just for fun, here is a cowboy who loads his ammunition with Black Powder. He has to shoot kind of slow because the smoke obscures his targets. You will notice he shifts his feet while shooting his rifle in order to see the targets past the smoke. And he shoots his Colts one handed. They do recoil a bit, so a little bit of time is lost bringing the pistols down from recoil for every shot. Actually, this guy does not care how fast he shoots, he just wants to hit all the targets.