Trim greens from turnips, discard any discolored leaves, and trim tough stems. Cut greens into 3-in.-long pieces (you should have about 5 heaping cups) and put in a medium nonreactive bowl. Add 1/2 tsp. salt and massage into greens until evenly distributed.
Peel turnips and thinly slice on a hand-held slicer. Put in a large nonreactive bowl, add 1 tsp. salt, and mix until evenly distributed. Set greens and turnips aside at room temperature 1 hour for salt to pull water from vegetables, massaging them once or twice. Greens will turn dark green and sliced turnips will become wilted and slightly translucent.
Pour 6 cups cold water and plenty of ice over sliced turnips and set aside 10 minutes to crisp. (No need to ice greens.)
Meanwhile, stir vinegar, sugar, jelly, and remaining 1/4 tsp. salt in a medium nonreactive bowl until sugar dissolves. Squeeze out as much excess liquid from greens as you can. Coarsely chop greens and add to vinegar mixture.
Lift ice from turnips, then drain turnips and squeeze out as much liquid as you can. Add turnips to bowl of greens and toss until well combined. Serve cold.
*Find turnips with greens at well-stocked grocery stores and farmers' markets. Or buy trimmed turnips plus 3 oz. (5 heaping cups) bagged turnip greens or mustard greens.