Melt butter in a large skillet over medium heat. Cook and stir turnips, radishes, garlic, and green onions in melted butter until hot, about 3 minutes. Stir carrots, red potatoes, and vegetable stock into the turnip mixture; bring to a simmer, add bay leaf, and reduce heat to low. Cook at a simmer, stirring occasionally, for 45 minutes.
Remove and discard bay leaf. Working in batches, pour the turnip mixture into a blender, filling the pitcher no more than half full. Carefully start the blender, using a few quick pulses to get the mixture moving before leaving it on to puree. Transfer to a large saucepan over low heat.
Stir milk, Worcestershire sauce, curry powder, brown sugar, and black pepper into the soup; cook and stir constantly until color and spices are evenly mixed. Let soup simmer over very low heat for another 10 minutes.