Start by adding the sugar and basil leaves to your saucepan. Use the back of a wooden spoon to press and muddle them together for a minute or two. This step bruises the basil leaves so they release their oils into the sugar, which is really where all that flavor comes from before you even turn on the stove.
Pour in 2 cups of water and set the pan over medium heat. Stir often so the sugar doesn't scorch on the bottom, and let it cook just until every bit of sugar has fully dissolved. You're not trying to boil it hard, just warm it enough to melt the sugar into the water completely.
Take the pan off the heat once the sugar has dissolved. Let the basil sit right in the warm syrup as it cools down to room temperature. This resting time is what really pulls the herbal flavor out, so don't rush it or skip it, even though it's tempting.
Once the syrup has cooled, pour it through a fine mesh strainer into your pitcher to remove the basil leaves. Add the remaining 4 cups of water along with your freshly squeezed lemon juice, then stir everything together well so the syrup doesn't settle at the bottom.
Pour your basil lemonade over a glass full of ice. Drop in a few lemon slices and a basil leaf or two for garnish, and it's ready to drink right away. If you have time to let the pitcher chill in the fridge for an hour first, it tastes even better ice cold.