Use a tec suit to tame. Put element in your inventory. Tec suit let's you breath underwater and can boost you through the water at high speed to get fish. I used a crossbow to kill sabertooth, then boost quickly to hesperonis to feed.
I killed the hesperonis I did not want to tame as these will kill the fish and make it difficult to find them.
What I did was use my Ichthyornis to Hunt fishes and bring the dead bodies for me.
Change his settings to "Hunt and retrieve", put him on your shoulder then you go swim. While you are swimming aim at The fish you want and double tap F (on PC). Ichthyornis will fly, kill the fish and bring the body to you.
Then you take the body to Hesperornis and feed it. Once you get used to it it's very fast and easy. I recommend following Hesp from behind so you dont need to adjust your aim all the time.
The most frustrating thing about taming these guys is not only do they take a lot of fish but the same button to feed them is the same button to drop the fish(i don’t know if it’s different in pc or ps4)and it’s constantly swinging around your face making it annoying because you’re constantly dropping it. Either make it so the controls to feed them are different then dropping it or make ur so it’s not constantly in ur face.
Tame to this kill fish and drag there body to it and once tamed if you feed them more dead fish corpses it will build up and lay a golden egg which is used as a buff to certain dinos or oil or a player buff
BIGGEST PAIN TO TAME EVER! have to get sabertooth fish kill them BUT do NOT harvest them you have to drag the body to it and feed it. Also good luck feeding it while the body is flailing everywhere in front of you. Also you CAN breed them.