For the filling: Preheat the oven to 375 degrees F.
Add the strawberries, granulated sugar, cornstarch and lime zest and juice to a bowl. Toss to combine and set aside.
For the crust and crumble: In a food processor, combine the flour, butter, brown sugar and baking powder. Pulse until a loose crumb forms; continue to pulse as you add the cream.
Grease a 9-by-13-inch pan with butter. Dump two-thirds of the flour mixture onto the prepared pan and press flat. Top with the strawberry mixture, spreading it into an even layer. Sprinkle over the remaining flour mixture, followed by the turbinado sugar.
Bake the bars until the top is golden and the strawberry layer is thick and bubbling, 28 to 30 minutes. Remove and allow to cool.
For the icing: Whisk together the powdered sugar and lime juice until smooth.
Cut the cobbler into small squares and drizzle with the icing. Add to a platter and serve.