1. Warm olive oil in a large skillet over medium-high heat. Add garlic and cook, stirring, until fragrant, about 30 seconds. Add mushrooms, oregano and 1/2 tsp. salt and cook, stirring, until mushrooms release their water, 3 to 5 minutes. Add turkey and cook, stirring and breaking up with a spoon, until no longer pink, about 8 minutes. Stir in tomatoes and sugar, bring to a simmer, and cook until thickened, 15 to 20 minutes. Season with salt and pepper; keep warm.
2. Bring 4 1/2 cups water and 1 tsp. salt to a boil in a small saucepan. Whisk in polenta and cook, whisking constantly, until thickened. Divide polenta among bowls, top with sauce, sprinkle with Parmesan and serve immediately.