If you can't find ripe mangoes, keep unripe ones unrefrigerated in a paper bag for a few days.
Combine mangoes, lemon juice, and 2 tablespoons of the sugar in a nonreactive bowl, and let sit for 2 hours to draw out the juices. Stir occasionally.
Boil together the water and remaining sugar until sugar dissolves. Let cool.
Puree the mango mixture in a food processor, then pass it through a strainer, pressing with the back of a spoon to remove any fibers. Combine with the sugar syrup. Chill, then pour into an ice-cream maker and freeze according to manufacturer's instructions.