Over medium heat, heat the oil in a large skillet. Add the onion, mushrooms, and salt and cook, stirring occasionally until the onions are soft and the mushrooms have shrunk and released their moisture, about 8 minutes
Add the garlic and chili powder and cook until fragrant, about 30 seconds. Add the ground turkey and break up the meat using a wooden spoon. Cook for about 4-5 minutes until the ground turkey is just about cooked through.
Stir in the pepper, brown sugar, tomato puree, ketchup, water, Worcestershire sauce, and hot sauce. Simmer until the sauce thickens and it's no longer runny about 10 minutes. Serve immediately on hamburger buns.