To tame a Fjordhawk, you must let it eat creatures you kill (wild or tamed). The taming increase varies depending on the creature it eats.
Based on Dododex's testing of dozens of creatures, it generally prefers larger creatures such as the Brontosaurus, Diplodocus, or Parcer (all three were equal), however the size is not always correlated to how effective the creature is. The Ovis is the most effective creature to feed it. The Phiomia tames it very well in comparison to other easy-to-hunt dinos (even outperforming the Allosaurus and the Therizinosaurus).
Note that the level of the creature it eats does not matter (a level 1 will increase the taming bar just as much as a level 150).
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When you are travelling between beds, move into an open area to wait for your fjordhawk’s arrival. This will keep the hawk from dropping your bag into tough-to-get-to places (building roofs, tek generators, tek troughs, etc ). I had to break ceilings and a cliff platform to get my bag recently.
In order to tame you have to let it follow you while you kill creatures. It will then proceed to eat the corpse, the taming bar raises with each bite it takes. Depending on what you kill determines how fast the taming bar rises. Phiomas worked well because it takes several bites. Keep your eye on it so it doesn't fly off and be careful not let it take damage or you will have to start over again.
The Fjordhawk is a scavenging bird in ARK: Survival Evolved that is included in the free DLC map, Fjordur, on June 12, 2022. It can fetch items and locate injured targets. After dying, the Fjordhawk can retrieve and return your items once you die.