Heat olive oil in a pot over medium heat. Cook and stir onion and garlic in hot oil until fragrant, about 2 minutes. Add carrots, celery, and red bell pepper; cook and stir until carrots begin to soften, 8 to 10 minutes.
Pour turkey stock into vegetable mixture and bring to a boil. Reduce heat to low and simmer until flavors blend, about 20 minutes.
Stir cooked turkey, lemon juice, and lemon zest into turkey stock mixture; simmer until heated through, about 5 minutes. Add spinach leaves, salt, and black pepper; simmer until spinach wilts and turns bright green, 1 to 2 minutes.