Heat olive oil in a stockpot over medium-high heat. Cook and stir turkey, onion, and garlic in the hot pot until browned and crumbly, 5 to 7 minutes. Add kale and cook until wilted, about 2 minutes.

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Stir in brown sugar, chili powder, cumin, oregano, onion powder, smoked and regular paprika, thyme, salt, pepper, and cayenne. Add chicken broth, crushed tomatoes, brown rice, and water; bring to a simmer.

Reduce heat to medium-low and cook until rice is tender, about 40 minutes.