HEAT 1 teaspoon of the olive oil in a large nonstick skillet over medium heat. Add the bell pepper and cook, stirring, until tender, 2 to 3 minutes.
IN a large bowl, combine the bell pepper, salmon, mayonnaise, cumin, yellow onion, garlic powder, paprika, oregano, cayenne (if using), and salt and black pepper to taste. Stir until combined. Shape the fish mixture into nine ¼-inch-thick patties, using a scant ¼ cup per patty.
ADD the remaining 3 teaspoons oil to the same skillet and heat over medium heat. Cook the patties, four or five at a time, in the hot oil, turning once, until browned on both sides, 7 to 9 minutes.
MEANWHILE, in a medium bowl, combine the mango, avocado, red onion, cilantro, jalapeño (if using), and lime juice. Serve the patties topped with some of the relish.