Heat olive oil in a large skillet over medium-high heat; stir in ground turkey. Cook and stir until turkey is crumbly, evenly browned, and no longer pink. Drain and discard any excess grease.

Stir in onion, bell peppers, and jalapeños; cook and stir until onion softened and translucent, about 8 minutes. Add garlic, cumin, chili powder, sugar, salt, black pepper, and thyme. Continue to cook until fragrant, about 8 more minutes, stirring occasionally.

Stir in 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.