In a medium saucepan, warm oil over medium-high heat. Add onion and cook, stirring often, until softened, 5 minutes. Add turkey and cook, breaking up large chunks and stirring often, about 5 minutes.
Pour in tomato sauce, barbecue sauce, brown sugar and 1/2 cup water. (Add extra brown sugar to sweeten sauce, if desired.) Season with salt and pepper and cook until mixture is slightly thickened, about 20 minutes.
Spoon mixture over 4 toasted or warmed hamburger buns and top with fixings.