Assume $\gamma$ is smooth and it has two points $p_1$ and $p_2$ with different curvatures, say $\kappa_1>\kappa_2$. Then one can touch $\gamma$ at $p_2$ from inside by a $g(\gamma)$ such that $g(p_1)=p_2$.
Since $\gamma$ and $g(\gamma)$ bound the same area, they intersect at some other points, in fact at least 2. By moving $g(\gamma)$ slightly, you can make 4 points of intersection.