For the salsa, mix together papaya, mango, bell pepper, red onion, cilantro, and lime juice. Let the mixture sit at room temperature for at least 30 minutes.
Preheat the grill to medium heat.
To make the burgers, combine 1/2 cup of the red onion, breadcrumbs, bell pepper, jerk sauce, and egg white in a large bowl. Stir until well combined, then add the ground turkey and mix thoroughly. Shape the mixture into 4 equal patties, each about 1 inch thick. Cover the patties and refrigerate for 20 minutes.
Lightly spray the patties with cooking spray and place them on a grill rack that has also been coated with cooking spray. Grill the patties for about 7 minutes per side or until they are fully cooked. Serve on Kaiser rolls or hamburger buns with the papaya-mango salsa on top.