Using potato masher, mash sugar, half of lime slices, and half of strawberries in large bowl until fruit is thoroughly crushed and sugar is completely wet, about 1 minute.
Add water and lemon juice and whisk until sugar is completely dissolved, about 1 minute. Strain mixture through fine-mesh strainer set over large bowl or pitcher, pressing on solids to extract as much juice as possible. Discard solids.
Add remaining lime slices and remaining strawberries to strained lemonade and chill for at least 1 hour. Serve.