The regular Turán 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) Turán 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.