Combine the sugar with 1 cup of the water in a small saucepan. Warm over medium heat for 2 to 3 minutes, stirring until the sugar completely dissolves; do not let the mixture boil vigorously. Set the syrup aside to cool briefly.

Place the cubed watermelon in a blender and process until completely smooth, usually 30 to 60 seconds. Work in batches if your blender is small.

For smoother lemonade, pour the watermelon purée through a fine-mesh strainer into a large bowl or pitcher, pressing gently on the solids. Skip straining if you prefer a little natural watermelon pulp.

Add the watermelon juice, fresh lemon juice, cooled syrup, and the remaining 3 cups of cold water to a large pitcher. Stir thoroughly so the sweet syrup and citrus are evenly distributed.

Taste the lemonade and adjust with additional lemon juice, sugar, or water as needed. The finished drink should taste fruity first, with a pleasant lemon tang and enough sweetness to remain refreshing.

Refrigerate the pitcher for at least 30 minutes. Serve over ice in individual glasses and finish with lemon slices or mint if desired.