Heat the olive oil in a large pot over medium heat.
Soften the onion and bell pepper with a pinch of salt, stirring, about 5 minutes.
Stir in the garlic and tomato paste. Cook until fragrant and darkened, about 1 minute.
Add the ground turkey. Cook, breaking it up with a spoon, until no longer pink, 5 to 7 minutes.
Sprinkle in chili powder, cumin, smoked paprika, oregano, coriander, salt, pepper, and cayenne. Toast 30 seconds.
Pour in diced tomatoes with juices, crushed tomatoes, and broth. Add both beans and stir well.
Bring to a gentle simmer. Reduce heat and cook uncovered, stirring occasionally, 20 to 25 minutes.
Finish with lime juice and brown sugar. Taste and adjust salt or heat as needed.
Serve hot with cilantro, shredded cheese, and a dollop of sour cream.