In a glass measuring cup, combine sugar with 1/4 cup water. Microwave until simmering, then stir until sugar dissolves. Put measuring cup in a bowl of ice and cold water and chill syrup until cold, about 10 minutes.
In a bowl, whisk 3 tbsp. syrup with the yogurt. Spoon into 6 (1/3 cup) popsicle molds*, filling each about halfway.
Put strawberries in a food processor. Pour in 3 to 4 tbsp. of remaining syrup and the rosewater (if using). Process until very smooth.
Pour strawberry purée on top of yogurt in molds. Freeze until slushy, about 1 hour (don't insert sticks yet).
Run a thin knife down and around inside edge of molds to create a blizzardlike effect. Insert sticks and freeze until solid, at least 5 more hours.
To unmold popsicles, either let sit at room temperature 5 minutes or run molds under warm water 10 to 15 seconds.
*We used Koji Ice popsicle molds ($13/set of 6; target.com).
Make ahead: Up to 2 weeks, frozen.