Make the Crust: Preheat oven to 325 degrees F. Line an 8-inch square baking dish with aluminum foil, allowing excess to hang over the edges of the pan; lightly grease the foil.
In a medium bowl, stir together the graham cracker crumbs, brown sugar, flour and salt to combine. Drizzle the melted butter over the mixture and use a fork to mix it until the entire mixture is moistened. Turn the mixture out into the prepared pan and press into an even layer. Bake until the crust to starts to brown, 12 to 15 minutes. Remove from oven and set aside.
Meanwhile, with an electric mixer on medium-high speed, beat the cream cheese until smooth, about 2 minutes. Add the granulated sugar and beat until it is completely incorporated. Add the eggs one at a time, mixing after each addition until well combined. Add the sour cream and vanilla extract and mix until incorporated. Pour the batter over the baked crust.
Using a spoon, drop dollops of the strawberry jam all over the surface of the cheesecake batter, then use a toothpick or skewer to gently swirl the jam into the batter. Bake until the edges are set but the center still jiggles slightly, 35 to 40 minutes. Allow the cheesecake to cool completely on a wire rack, about 2 hours. Cover and refrigerate until thoroughly chilled, at least 3 hours. Lift the cheesecake out of the pan and slice into bars. Leftovers can be stored in the refrigerator, in an airtight container, for up to 3 days.