Whisk soy sauce, rice vinegar, lime juice, brown sugar, sesame oil, and cornstarch together in a small bowl; set aside.

Heat 1 tablespoon oil in a large skillet over medium heat. Add the mushrooms, red and green bell peppers, and onion, and cook just until vegetables are tender, 5 to 6 minutes. Remove to a plate.

Add the remaining tablespoon oil to the skillet. Add ground turkey and cook, breaking up the meat into chunks with a spatula, until lightly browned and cooked through, 5 to 6 minutes. Stir in garlic and cook until fragrant, 30 seconds to 1 minute.

Reduce heat to medium-low, return vegetables to the skillet, give sauce a good stir, and pour over meat and veggies. Stir until everything is well coated, and simmer until the sauce is thickened slightly, about 2 minutes.