Heat olive oil in a large skillet over medium heat. Add the onion and bell pepper, cooking for 5 minutes until softened.
Add the garlic and cook for another minute until fragrant.
Stir in the chopped turkey and cook for 2-3 minutes to warm through.
Add ketchup, brown sugar, Worcestershire sauce, mustard, water, salt, and pepper. Stir everything together until well combined.
Reduce heat to medium-low and simmer for 8-10 minutes, stirring occasionally, until the sauce has thickened to your liking.
Toast the hamburger buns if desired, then pile on the turkey mixture and serve immediately. Don't forget the napkins!