Heat oil in a large soup pot over medium heat. Add ground turkey, season with salt and pepper, and brown until it's no longer pink, breaking it up with a spatula as it cooks.
Add the onion, bell pepper, and spices. Cook for 1-2 minutes or until the onion becomes translucent. Toss in the sweet potatoes and brown a little so they will become nice and flavorful.
Add the broth, 1 cup of water, and 1 teaspoon salt. Bring to a simmer and cook until the sweet potatoes are fork-tender.
Add the kale and continue to simmer until it's wilted.