Heat a large skillet over medium-high heat. Add ground turkey and cook, breaking it up with a wooden spoon, until browned and cooked through, about 6-7 minutes.
Add the diced onion and bell pepper to the skillet. Cook for 4-5 minutes until softened, then add garlic and cook for another minute until fragrant.
Stir in tomato sauce, tomato paste, brown sugar, Worcestershire sauce, vinegar, Dijon mustard, and smoked paprika. Mix well to combine.
Reduce heat to medium-low and simmer for 8-10 minutes, stirring occasionally, until the mixture thickens. Season with salt and pepper to taste.
Toast the hamburger buns if desired, then spoon the sloppy joe mixture onto the bottom buns (about 2/3 cup per serving). Top with the other bun half and serve immediately.