Heat the oven to 350°F (175°C). Line an 8-inch or 9-inch square pan with parchment paper, leaving enough overhang to lift out the chilled bars.

Stir the graham-cracker crumbs, 1/4 cup sugar, and melted butter together. Press the mixture firmly and evenly into the pan, then bake for 8–10 minutes and cool briefly.

Cook the strawberries, 2 tablespoons sugar, and lemon juice in a small saucepan over medium heat until softened. Stir together the cornstarch and water, add the slurry, and cook for 1–2 minutes until thickened; mash or purée and set aside.

Beat the softened cream cheese and 1/2 cup sugar until smooth. Add the eggs one at a time, mixing just until incorporated, then blend in the sour cream, vanilla, and flour.

Pour the cheesecake batter over the crust. Spoon small amounts of strawberry mixture over the surface and gently drag a knife or skewer through it to create a swirl.

Bake for 35–40 minutes, until the edges are set and the center has only a slight jiggle. Cool completely at room temperature for about 1 hour, then refrigerate for at least 2 hours before cutting.