Blend strawberries, water, rum, and limeade concentrate together in a blender until the berries are mostly pureed.
Pour about half the strawberry mixture into another container.
Add ice 1/4 cup at a time to the blender and blend until the ice is completely crushed. Continue adding ice and blending until the consistency is that of a milkshake. Pour finished daiquiri into 2 glasses and repeat blending ice with the reserved strawberry mixture. Garnish glasses with lime wedges.