In a small saucepan, combine ½ cup water and granulated sugar. Bring to a boil over medium heat, stirring until sugar dissolves. Remove from heat and let cool.

Cut fresh watermelon into cubes and puree in a blender until smooth. Strain through a fine mesh sieve to remove pulp.

In a large pitcher, combine the watermelon puree, cooled simple syrup, lemon juice, and cold water. Stir gently to combine.

Refrigerate the pitcher for at least an hour. Serve over ice, garnished with a watermelon wedge or mint if desired.