  The regular Tur\'an number of a graph $F$, denoted by rex$(n,F)$, is the
largest number of edges in a regular graph $G$ of order $n$ such that $G$ does
not contain subgraphs isomorphic to $F$. Giving a partial answer to a recent
problem raised by Gerbner et al. [arXiv:1909.04980] we prove that rex$(n,F)$
asymptotically equals the (classical) Tur\'an number whenever the chromatic
number of $F$ is at least four; but it is substantially different for some
3-chromatic graphs $F$ if $n$ is odd.
