  In 1939 P\'al Tur\'an and J\'anos Er\H{o}d initiated the study of lower
estimations of maximum norms of derivatives of polynomials, in terms of the
maximum norms of the polynomials themselves, on convex domains of the complex
plane. As a matter of normalization they considered the family
$\mathcal{P}_n(K)$ of degree $n$ polynomials with all zeros lying in the given
convex, compact subset $K\Subset {\mathbb C}$. While Tur\'an obtained the first
results for the interval $I:=[-1,1]$ and the disk $D:=\{ z\in {\mathbb C}~:~
|z|\le 1\}$, Er\H{o}d extended investigations to other compact convex domains,
too. The order of the optimal constant was found to be $\sqrt{n}$ for $I$ and
$n$ for $D$. It took until 2006 to clarify that all compact convex
\emph{domains} (with nonempty interior), follow the pattern of the disk, and
admit an order $n$ inequality.
  For $L^q(\partial K)$ norms with any $1\le q <\infty$ we obtained order $n$
results for various classes of domains. Further, in the generality of all
convex, compact domains we could show a $c n/\log n$ lower bound together with
an $O(n)$ upper bound for the optimal constant. Also, we conjectured that all
compact convex domains admit an order $n$ Tur\'an type inequality. Here we
prove this for all \emph{polygonal} convex domains and any $0< q <\infty$.
