Make the peach puree:

In a blender, combine the sliced peaches with 1/2 cup of water. Blend until smooth. Strain the puree through a fine mesh strainer into a bowl to remove pulp. Set aside.

Prepare the lemon syrup:

In a medium saucepan, combine 3/4 cup of water with the sugar. Heat over medium heat, stirring occasionally, until the sugar is completely dissolved. Remove from heat and let cool.

Mix the lemonade:

In a large pitcher, combine the peach puree, lemon juice, cooled sugar syrup, and the remaining 4 cups of cold water. Stir well to mix.

Chill and serve:

Refrigerate the lemonade for at least 30 minutes to chill. Serve over ice, garnished with fresh mint leaves and peach slices if desired.