Coarsely grate (with a box grater or a food processor fitted with the grater attachment) potatoes and turnips, all scrubbed and trimmed. Squeeze to remove as much moisture as possible; transfer vegetables to a medium bowl. Toss with coarse salt and ground pepper.
Heat olive oil in a large skillet over medium heat. Form potato mixture into four tightly packed patties; place in skillet, flattening gently with a spatula to a 3/4-inch thickness. Cook patties, turning once, until browned and cooked through, about 10 minutes per side (reduce heat if patties start to brown too quickly, and add more oil to skillet if necessary). Transfer to paper towels; sprinkle with salt.