1. Preheat the oven to 350°F and line an 8×8-inch baking pan with parchment paper. Lightly grease the paper for easy removal.

2. In a saucepan over medium heat, combine chopped strawberries, strawberry jam, and lemon juice. Cook until slightly thickened, then allow the mixture to cool completely.

3. In a large bowl, mix melted butter, brown sugar, 1/4 cup granulated sugar, and 1 teaspoon vanilla extract until smooth.

4. Add flour, baking powder, and salt to the butter mixture and stir until a soft dough forms.

5. Press the dough evenly into the prepared baking pan to create the crust layer.

6. In another bowl, beat softened cream cheese until smooth and creamy.

7. Add 1/2 cup granulated sugar and mix well. Beat in eggs one at a time, then add the remaining vanilla extract.

8. Pour the cheesecake filling evenly over the crust layer.

9. Spoon the cooled strawberry mixture over the cheesecake filling and gently swirl using a knife or toothpick.

10. Bake for 35 to 45 minutes or until the edges are lightly golden and the center still has a slight jiggle.

11. Allow the bars to cool completely at room temperature before refrigerating for at least 4 hours.

12. Slice into squares and serve chilled.