Peel and slice the 3 peaches, discarding the pits, and add them to a medium saucepan. Add the 3/4 cup granulated sugar and 1 cup water to the same saucepan.
Stir everything together over medium heat until the sugar starts to dissolve. Bring the mixture to a gentle simmer, not a full boil, and reduce the heat slightly once it starts bubbling.
Simmer for 8 to 10 minutes, stirring every couple of minutes, until the peaches are very soft and starting to fall apart. Remove the saucepan from the heat once the peaches are fully softened.
Use a potato masher or the back of a wooden spoon to mash the peaches directly in the pan, breaking them down as much as possible. Set a fine mesh strainer or cheesecloth-lined strainer over a heatproof bowl or measuring cup.
Pour the peach mixture through the strainer, pressing the solids with the back of a spoon to push out as much syrup as possible. Discard the leftover peach pulp and skins, or save them to stir into oatmeal or yogurt.
Let the strained peach syrup sit at room temperature for about 15 to 20 minutes. Give it a stir partway through so it cools evenly and doesn't form a skin on top.
The syrup should be just warm or fully cool, not hot, before it goes into the pitcher. Pour the cooled peach syrup into a large pitcher.
Add the 1 cup of fresh lemon juice to the pitcher. Pour in the 4 cups of cold water.
Stir the mixture thoroughly with a long spoon until the syrup is fully combined and no streaks remain. Taste and adjust if needed—add a splash more water for a milder flavor, or a spoonful more syrup for a stronger peach taste.
Refrigerate the pitcher for at least 20 minutes if a colder drink is preferred, or serve right away over ice. Fill glasses with ice and pour the lemonade over the top.
Garnish with a lemon slice, a fresh peach slice, or a sprig of mint if desired. Stir the pitcher again before refilling glasses since the peach solids can settle toward the bottom.