Mix grated potatoes, turnips, and onion together in a bowl, draining off any excess liquid. Mix egg, parsley, basil, dill, salt, and pepper into potato mixture. Stir flour into potato mixture, adding enough flour to make a thick batter.

Heat enough oil to fill the bottom of a heavy skillet to 1/4-inch depth over medium-high heat.

Working in batches of 3 or 4 pancakes, spoon batter, 1/4 cup per pancake, into hot oil and flatten to make 1/2-inch thick pancakes. Fry pancakes in the hot oil until golden brown, 2 to 3 minutes per side. Transfer pancakes to a paper towel-lined plate.