Heat the olive oil in a skillet over medium heat. Add the turkey cubes and cook until lightly browned on all sides, about 5 minutes.
Transfer the browned turkey to the slow cooker. Add the onion, carrots, garlic, white beans, chicken broth, thyme, rosemary, salt, and pepper.
Stir to combine, then cover and cook on low for 6 hours.
Stir in the chopped kale during the last 30 minutes of cooking so it wilts but stays bright.
Taste and adjust salt and pepper before serving.