Heat olive oil in a large skillet over medium-high heat and stir in the ground turkey. Cook and stir until the turkey is crumbly, evenly browned, and no longer pink. Drain and discard any excess grease. Stir in the onion, green and red bell peppers, and jalapeno peppers. Cook and stir until the onion is softened and translucent, about 8 minutes. Add the garlic, cumin, chili powder, sugar, salt, black pepper, and thyme. Continue to cook until fragrant, about 8 more minutes, stirring occasionally.
Stir in the beef broth, pinto beans, red beans, diced tomatoes, tomato sauce, and tomato paste. Bring to a boil, then reduce heat to low. Simmer until thickened, about 45 minutes, stirring occasionally. For a thinner chili, cover the pot when simmering.
