  In 1939 Tur\'{a}n raised the question about lower estimations of the maximum
norm of the derivatives of a polynomial $p$ of maximum norm $1$ on the compact
set $K$ of the complex plain under the normalization condition that the zeroes
of $p$ in question all lie in $K$. Tur\'{a}n studied the problem for the
interval $I=[-1,1]$ and the unit disk $D$ and found that with $n$ denoting the
degree of $p$ and with $n$ tending to infinity, the precise growth order of the
minimal possible derivative norm (oscillation order) is $\sqrt{n}$ for $I$ and
$n$ for $D$. Er\H{o}d continued the work of Tur\'{a}n considering other
domains. Finally, in 2006, Hal\'{a}sz and R\'{e}v\'{e}sz proved that the growth
of the minimal possible maximal norm of the derivative is of order $n$ for all
compact convex domains.
  Although Tur\'{a}n himself gave comments about the above oscillation question
in $L^q$ norms, till recently results were known only for $D$ and $I$.
Recently, we have found order $n$ lower estimations for several general classes
of compact convex domains, and proved that in $L^q$ norm the oscillation order
is at least $n/\log n$ for all compact convex domains. In the present paper we
prove that the oscillation order is not greater than $n$ for all compact (not
necessarily convex) domains $K$ and $L^q$-norm with respect to any measure
supported on more than two points on $K$.
