Combine strawberries, sugar and 1 cup water in a glass bowl. Let sit at room temperature for 1 to 4 hours, until berries infuse the sugar with their juice. Puree strawberry mixture until smooth.

Pour mixture through a mesh sieve into a pitcher, using a wooden spoon to push the puree through the sieve. Discard seeds and pulp. Stir lime juice and remaining 2 cups water into the strawberry puree; stir and serve.