Heat olive oil in a large pot over medium heat, then add the onion, garlic, bell pepper, and carrot. Cook for 4–5 minutes, stirring frequently, until the onion is translucent and vegetables are softened.
Add the ground turkey to the pot and break it into small crumbles with a wooden spoon. Cook for 5–7 minutes, stirring occasionally, until no pink remains.
Stir in the black beans, kidney beans, diced tomatoes with their juices, corn kernels, and chicken broth until everything is evenly combined.
Sprinkle in the chili powder, cumin, smoked paprika, salt, black pepper, and cayenne (if using), then stir well and scrape up any browned bits from the bottom of the pot.
Bring to a boil over medium-high heat, then reduce to low, partially cover, and simmer for 40–45 minutes—stirring every 10 minutes—until the chili thickens and the flavors meld.
Taste and adjust salt, pepper, or cayenne as needed, then ladle into bowls and top with your choice of avocado, Greek yogurt, shredded cheese, cilantro, and a squeeze of lime.