Make the basil simple syrup: Combine sugar and 1 cup water in a medium saucepan over medium heat. Stir until sugar completely dissolves, about 3-4 minutes. Remove from heat, add the basil leaves, and gently muddle them with a wooden spoon to release their oils. Cover and let steep for 10 minutes.
Prepare your lemons: While the syrup steeps, roll your lemons firmly on the countertop using the palm of your hand to help release more juice. Cut in half and juice them, being careful to remove any seeds.
Strain the basil syrup: Pour the syrup through a fine-mesh strainer into your pitcher, pressing gently on the basil leaves to extract all the flavor. Discard the used basil leaves.
Combine all ingredients: Add the fresh lemon juice to the pitcher with the basil syrup. Pour in the remaining 4 cups of cold water and stir thoroughly to combine all flavors.
Chill and serve: Refrigerate until cold, at least 1 hour for best flavor. Serve over ice, garnished with fresh basil leaves and lemon slices.