Blend the strawberries in a food processor or blender until smooth, then strain the puree through a fine-mesh sieve if you prefer a seedless result. In a small saucepan set over medium heat, combine the sugar with 2 cups of water, bringing the mixture to a boil while stirring until the sugar fully dissolves. Take off the heat and allow the syrup to reach room temperature. Pour the freshly squeezed lemon juice through a fine sieve into a 2-quart pitcher, add the cooled syrup and strawberry puree, and mix thoroughly. Refrigerate until well chilled. Just before serving, pour in 2 1/2 cups of cold water and stir; adjust with additional water to reach your desired sweetness. Serve generously over ice.