Igor Pak's argument takes care of the convex case. So it remains to consider the nonconvex case. To this end it is enough to note that if $\gamma$ is not convex, then it must intersect some line $L$ at least $4$ times. Let $g(\gamma)$ be the reflection of $\gamma$ with respect to $L$. Then $\gamma$ and $g(\gamma)$ intersect at least $4$ times.

Igor Pak's argument takes care of the general convex case. So to complete the proof (without assuming any smoothness) it remains to consider the nonconvex case. To this end it is enough to note that if $\gamma$ is not convex, then it must intersect some line $L$ at least $4$ times. Let $g(\gamma)$ be the reflection of $\gamma$ with respect to $L$. Then $\gamma$ and $g(\gamma)$ intersect at least $4$ times.