By the definition of the convolution, $$ |h(x)| \lesssim \int_{-\infty}^{\infty} \exp\left(-k|y|^{\gamma}-\frac{(x-y)^2}{2}\right)\, dy . $$ We can now split this into two parts: $|x-y|>|x|^{1/2}$ and $|x-y|\le |x|^{1/2}$. Then first integral is $O(e^{-|x|/2})$, and the second one is $O(e^{-c|x|^{\gamma}})$ (because $|y|\gtrsim |x|$ here for large $x$).
This can of course be done more carefully if you want optimized bounds.