Heat the sugar and water in a small saucepan over medium-high heat, stirring until the sugar has dissolved. Remove from heat and set aside to cool slightly. (Alternatively, place the sugar and water in a container with a tight-fitting lid. Shake vigorously for 3 minutes. Let rest for about 1 minute. Shake again until all the sugar has dissolved, about 30 seconds more.)
Place the raspberries and lemon juice in the bowl of a food processor fitted with a blade attachment and process until smooth, about 1 minute.
Set a fine-mesh strainer over a medium bowl and strain the mixture, pressing down on the solids and scraping the inside of the strainer with a rubber spatula until mostly seeds remain (you should have about 2 1/3 cups raspberry purée); discard the solids.
Whisk the sugar syrup into the raspberry purée. Cover and refrigerate until cold, at least 1 hour.
Whisk the raspberry mixture to recombine. Freeze in an ice cream maker according to the manufacturer’s instructions.
Transfer to a container with a tight-fitting lid and place in the freezer until completely frozen, at least 3 hours. Let sit at room temperature to soften for a few minutes before serving.