Heat the oil in a medium skillet over medium-high heat. Add the onion and cook, stirring, until slightly soft, about 4 minutes.
Add the turkey and cook, crumbling with the back of a wooden spoon, until no trace of pink remains, 5 to 7 minutes.
Add the tomatoes and their juices, beans, bell pepper, tomato paste, 1½ cups water, chili powder, and 1¼ teaspoons salt. Bring to a boil.
Reduce heat and simmer until slightly thickened, about 15 minutes.
Divide among bowls and top with the sour cream, Cheddar, and onion, if using.