Heat the oil in a large skillet over medium-high heat. Add the turkey and cook, breaking it up with a spoon, until browned, 3 to 5 minutes.
Add the fish sauce, sugar, and ¼ cup water to the skillet and cook, stirring, until the liquid has almost evaporated, 2 to 4 minutes.
Meanwhile, in a large bowl, toss the cabbage, cucumber, mint, and peanuts with the lime juice. Top with the turkey mixture and serve with lime wedges.