The trick is to argo one, have him chase you into an underwater cave where he will not be able to fit through the door but his long neck and head does. when you do this he will not de-agro, you can then get a clear shot from the cave with a bow and tranq arrows
There is a glitch at the redwood on the center, where plesiosaurs will spawn really near the coast and whole groups of 2 or 3 are glitched in the water, meaning you can knock out multiple plesiosaurs at the time.
My trusty sarcosuchus made knocking my plesi exceedingly simple, for three reasons. The sarco is fast, it ascends quickly, and you can shoot from its back. Once I separated my plesiosaur from her friend and killed it, the procedure was simple. When the plesi reaches a certain height it begins to flee. I dragged her up then filled her with tranqs until she aggro'd again. Repeat process until unconscious. Simple with sarco.
Taming one is not too difficult. A high level won’t take too many tranq to knock out. When it’s knocked out, feeding it a soothing balm will allow u to tame it quickly, even with regular raw meat. Don’t forget to separate the plesiosaur u want from others in the area. Good luck!
They can't fit through the resource caves' entrances. Just lure it into one and pull out your crossbow/harpoon gun for the knock out. Just be aware of any nearby megalodons, as they can fit into the cave easily and ruin your swimming session.