Separate stems from kohlrabi bulb, trim, and discard tough bottoms of stems. Half leaves lengthwise then thinly shred crosswise. Trim root end from bulb and peel away tough outer layer; halve lengthwise.
Fit a food processor with a shredding blade (or use a box grater) and shred kohlrabi bulb and turnip.
In a medium bowl, whisk together lime juice, peanut oil, honey, and sesame oil; season with salt and pepper. Add scallions, kohlrabi leaves and bulb, and turnip to bowl; toss to coat. Let stand at least 15 minutes.