Bring broth to a boil in a Dutch oven. Stir in lentils; return to a boil. Reduce heat and simmer, covered, until tender, 20 to 25 minutes. Drain in a colander.
Heat oil in the Dutch oven over medium heat. Add onion and carrot; cook until tender, 5 to 6 minutes. Add turkey and garlic; cook, stirring to break up lumps, until turkey is browned. Drain any grease.
Stir in tomatoes, olives, chili powder, tomato paste, and vinegar. Bring to a boil; reduce heat and simmer, stirring frequently, until slightly thickened, about 8 minutes. Stir in cooked lentils; cook until heated through, about 2 minutes more.
