Island 4 is a two-boss chain. The Albatross is the story boss, but it only appears when you lay a dead Tuna on the grass, and the Tuna has to be caught first with a $1,200 lure. Bring the Albatross Head to the person inside the shop for the coordinates of the volcano.
Gear: one Sniper Rifle
The gun store is up the stairs from the dock and to the left. The Sniper Rifle ($3,800) is the only weapon you need here. Add the Suppressor, the Laser Sight and the Extended Mag, and buy every bullet-damage tier the store sells. The red dot sight and compensator can be skipped; the iron sights are good enough. A scope is optional: it makes the bird easier to follow when it circles high, but it slows down quick shots from cover.
That kit costs far more than the rifle alone, so most players farm here first.
Farm money with the Professional Lure
Buy Professional Lures at the bait shop. They bring up Rocks catches such as Bass, Halibut and Tigerfish worth $250 to $310 each before cooking, and the grill on this island still gives the 1.5x bonus.
A known trick multiplies that further: fish from the small shack and carry each live catch into its back right corner. The creature tries to swim through the wall and stays pinned there, which gives you time for trick kills. A full 360-degree spin while jumping before the killing shot stacks kill-score bonuses up to about 7x, and cooking afterwards multiplies the result by 1.5 again. The same spins count toward the Impressive and 360 no scope achievements.
Before the boss fights, take the rifle outside and practise on the birds flying over the island.
Catch and kill the Tuna
Buy the Professional Boss Lure ($1,200) at the bait shop, fit it on the Fishing Rod and cast from the shore. The Tuna has 5,000 health on Normal and was nerfed slightly in patch 1.0.12.

The Tuna aims its jumps at where you are heading. Run to one side for half a second, then cut back the other way, and it lands where you would have been. It is most dangerous when it stays on the ground instead of jumping; keep circling, and sprint away if you have to. Shoot in the pauses between charges.
If the Tuna keeps winning, there is a cheaper route. Drop a pile of Dynamite sticks on the floor of the gun shop, catch the Tuna right outside, pull it inside onto the pile, step back out and shoot the Dynamite with any gun. The blast kills it in one go.
Do not sell or cook the dead Tuna. A spare one sells for $2,000, but the one you need must stay whole.
Summon the Albatross
Carry the dead Tuna to the middle of the grassy area and drop it. Within a few seconds the Albatross swoops in, and you get a couple of free shots while it lands.

The Albatross has 7,800 health. Fight from the house next to the grass. Listen for its call: a loud screech means it is about to dive at you, so step inside under the roof. A quieter call means it is dropping eggs, which gives you time to step out and shoot.

It often circles behind the building. A red dot marks the bird even when it is out of sight, so track that dot from cover and shoot when it comes back into view rather than chasing it into the open. The walkway and bridge are also safe cover from the dive. In co-op, spread out behind different buildings: it targets one player at a time, and the others shoot freely.
The Dynamite trick works here too: ring the Tuna with sticks before you drop it, and shoot the pile when the Albatross lands on the bait. Timing is hard, so keep the house as your fallback.
Before patch 1.0.10, dying during the Tuna fight and then calling the bird could leave you unable to respawn. That bug is fixed.
Get the head, even if it lands in the sea
When the Albatross dies, its meat chunks scatter ($1,000 each, worth cooking) and the Albatross Head drops. The head is small and can land offshore. Do not swim for it, because dying in the water with the head is a real risk; take the boat out and pick it up from there.
Give the head to the person in the red shirt inside the shop. He thanks you and hands over a flash drive with the coordinates of Island 5, and the Terrorizing bird achievement unlocks.

Hold the Radar on the boat: the new magenta dot is the volcano island.
Next: Island 5, the Volcano, where the Assault Rifle and the final whale fights are.
Source video
Some numbers in this guide come only from player videos and are not officially verified.