Proving that $\iint f(x) g(x+y) dx dy \leq \int g^2(x) dx$

Let $D \subset \mathbb{R}^n$ be bounded, and $f,g: \mathbb{R}^n \rightarrow \mathbb{R}$. Furthermore let $f$ be nonnegative and such that $\int_D f dx= 1$.

I would like to prove that$$\int_D \int_D f(x)g^2(x+y)dxdy \leq \int_D g^2(y).$$

My attempt has been to use the fact that $\int_D f dx = 1$ to define a new measure $df = fdx$ which is now a probability measure, and then to use Jensen's inequality but that doesn't seem to get very far. The biggest obstacle for me has been going from $g(x+y)$ to $g(y)$ in the integral.

Any tips on how I can proceed?

EDIT: Based on the counter-example given by Matt Werenski and the comment given by Raghav, I will strengthen the assumptions by requiring $f \in C^\infty(D)$ and that the support of $f$ is strictly contained within the support of $g$.

EDIT 2: Based on more comments and the answer by Julio Puerta, it seems I need the support of $g$ to be contained in $D$, so I will assume this as well.