Heat the oil in a large saucepan over medium-high heat. Add the onion, bell pepper, and garlic. Cook, stirring occasionally, until tender, 4 to 6 minutes.
Add the turkey and cook, breaking up with a spoon, until no longer pink, 3 to 5 minutes.
Stir in the tomato paste, cumin, and chili pepper. Cook, stirring, for 1 minute.
Add the tomatoes and their juices, beans, ½ cup water, 1 ¼ teaspoons salt, and ¼ teaspoon pepper. Bring to a boil.
Reduce heat and simmer, stirring occasionally, until slightly thickened, 12 to 15 minutes.
Serve the chili with the sour cream and cilantro.