In a saucepan over high heat, mix together the water and the sugar. Cook until the sugar dissolves and it becomes a clear syrup. Remove from heat and set aside to cool while you do the next steps.
Add the strawberries and the 1 cup cold water to a blender. Pulse until the mixture is pureed.
To a half-gallon pitcher, add the lemon zest, lemon juice, 1 cup cold water, ice, and lemon and strawberry slices, if using. Stir in the strawberry puree and the simple syrup until well combined. Taste for sweetness and add more cold water if needed. Refrigerate to chill through or serve over ice.