In a large measuring cup, stir together the lemon juice and sugar until the sugar is dissolved. If you like, heat the lemon juice/sugar mixture for a minute or so in the microwave, or in a saucepan, to speed up the dissolving. Pour the lemon juice/sugar mixture into the pitcher of water and stir. Taste for sweetness. We like our lemonade on the sweet side, so we sometimes add a bit more sugar. Chill.
For one serving, fill a 12-ounce glass with 3 or 4 or 5 or 6 (or however many you like) big basil leaves and a slice of lemon. Muddle until fragrant. Fill glass with ice cubes. Add lemonade. Crush a few more small basil leaves in your fingers, and float the fragrant leaves on top. Sip.
To save some time, make a big pitcher of basil lemonade. Muddle a few handfuls of basil leaves, a few slices of lemon, and the turbinado sugar in a large pitcher. Add the water and lemon juice. Stir until sugar is dissolved. Taste for sweetness. Chill.
Alternatively, make a simple syrup by heating 1 ½ cups of the sugar and 1 ½ cups of the water, and let cool. In the pitcher, muddle the basil and lemon slices. Add the lemon juice, remaining water, and simple syrup. Chill. Serve on the rocks, and top with a few crushed basil leaves.