In a small saucepan, combine 1 cup water, the sugar and 2 to 3 mint sprigs. Bring to a simmer, stirring to dissolve sugar. Remove from heat, cover and let sit, steeping, 20 minutes. Discard mint sprigs.
Transfer the mint syrup to a large pitcher and stir in lemon juice and remaining 4 cups water. Add remaining mint sprigs; muddle by gently crushing them against the bottom of the pitcher 3 to 4 times with a wooden spoon. (Do not bruise or tear the leaves; it will make them taste bitter and grassy.)
Chill until cold, at least 4 hours. Remove and discard mint sprigs. Serve lemonade over ice garnished with additional fresh mint.