Melt butter in a saucepan over medium heat. Cook and stir garlic in melted butter until browned, about 5 minutes. Stir turnips into the butter and garlic; cook and stir until just beginning to soften, 7 to 10 minutes.
Add cream and Parmesan cheese to the turnip mixture; bring to a simmer and cook, stirring regularly, until the liquid thickens, 10 to 15 minutes. Stir green onion, oregano, parsley, salt, and black pepper into the turnip mixture; cook until the green onion slightly wilts, 1 to 2 minutes.
