Preheat the oven to 375°F (190°C).
Heat olive oil in a large skillet over medium heat.
Add diced onion and cook for 3-4 minutes until softened.
Add minced garlic and cook for 1 minute until fragrant.
Add ground turkey to the skillet and cook until browned, breaking it up with a spoon, about 6-8 minutes.
Add diced red bell pepper and sliced mushrooms to the skillet and cook for 5 minutes until vegetables are tender.
Stir in dried oregano, dried basil, salt, and black pepper.
Add the canned diced tomatoes with their juice and tomato sauce. Stir to combine and simmer for 5 minutes.
Remove the skillet from heat and stir in the cooked white rice.
Transfer the turkey mixture to a 9×13 inch casserole dish.
Sprinkle shredded mozzarella cheese evenly over the top.
Sprinkle shredded cheddar cheese over the mozzarella.
Sprinkle grated Parmesan cheese on top of all the cheese layers.
Bake uncovered in the preheated oven for 20-25 minutes, until the cheese is melted and bubbly.
Remove from oven and let the casserole rest for 5 minutes before serving.