1. Season the shrimp with salt and pepper and cook in 3 batches: For each batch, heat 1/3 cup of the olive oil in a large deep skillet. Add one-third of the shrimp and cook over high heat, turning once, until golden, about 3 minutes. Add 2 tablespoons of the lemon juice and 1 tablespoon of the dill and toss to coat, then transfer the shrimp as it's cooked to a large deep serving platter. Add 1/4 cup of the water to the skillet and boil over high heat, scraping up any browned bits from the bottom of the skillet, until the juices are reduced to 2 tablespoons; add to the cooked shrimp.
2. Add the remaining 2 tablespoons of lemon juice and 1 tablespoon of dill to the shrimp and season with salt and pepper. Toss well. Serve the shrimp warm or at room temperature, with the warm pita bread.
Make Ahead: The cooked shrimp can be refrigerated for up to 3 hours. Bring to room temperature before serving.