In a small saucepan, combine 1 cup of water and the granulated sugar. Heat over medium heat, stirring occasionally, until the sugar has completely dissolved to make a simple syrup. Remove from heat and let it cool.
Meanwhile, juice the lemons until you have 1 cup of fresh lemon juice.
In a large pitcher, combine the freshly squeezed lemon juice, the cooled simple syrup, and the remaining 4 cups of cold water. Stir well to mix.
Taste the lemonade and adjust the sweetness or tartness by adding more sugar or lemon juice, if desired.
Chill the lemonade in the refrigerator for at least 30 minutes before serving to allow the flavors to meld.
To serve, stir gently then fill glasses with ice cubes and pour the chilled lemonade over the ice. Garnish each glass with a slice of lemon and a sprig of fresh mint, if using.