Stir 4 oz. ginger, scrubbed, thinly sliced (about 1 cup), 1 cup (200 g) sugar, ½ lb. fresh strawberries, hulled, quartered, and ½ cup water in a medium saucepan until sugar is dissolved. Set over medium-high heat and bring to a simmer. Reduce heat; simmer gently until strawberries have lost most of their color, 15–20 minutes. Remove pan with strawberry syrup from heat.
Using a potato masher or fork, gently crush remaining ½ lb. fresh strawberries, hulled, quartered, in a medium bowl. Strain strawberry syrup through a fine-mesh sieve over crushed berries and press on cooked berries in sieve to extract as much liquid as possible; discard cooked berries and ginger. Add ½ cup fresh lime juice (from about 6 limes) and ¼ tsp. kosher salt to strawberry mixture and stir to combine. Chill until cold, about 1 hour.
To serve, divide strawberry mixture among 4 glasses filled with ice to come about halfway up sides; top off with club soda. Garnish each with a lime wedge and serve with straws if desired.