Heat olive oil in a Dutch oven or soup pot over medium-high heat. Add the ground turkey, salt, pepper, garlic powder, onion powder, rosemary, and thyme. Cook until the meat is browned and fully cooked.
Stir in the chopped spinach and potatoes. Cook for 2–3 minutes, just until the spinach begins to wilt.
Pour in the broth and milk. Bring the soup to a gentle boil, then reduce to medium heat. Let it simmer uncovered, stirring occasionally, until the potatoes are fork-tender (about 20–30 minutes).
Taste and adjust seasoning if needed. Ladle into bowls and enjoy with warm sourdough bread or a side salad.