For full Hesperornis taming details, get the taming calculator app.
To tame a Hesperornis, you must kill a fish, and drag the fish's corpse to the Hesperornis to feed it. The fish's size plays a major factor in how many fish will be required (a 1.0x fish appears to fill up the taming bar twice as much as a 0.5x fish). Level of fish does not appear to impact taming. The Sabertooth Salmon is by far the most effective fish, equivalent of about 4 Coels of the same size, while the Piranha is equivalent to about 2.4 Coels.
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.
When you anger it. Don’t try to laugh because you will laugh. It looks even more pathetic than a dodo rushing at you and it deals LITERALLY 1 damage. Upgrading it to 200% means it only deals like 2. Don’t take it to fight.