Kim and Vu made the following conjecture (\textit{Advances in Mathematics}, 2004): if $d\gg \log n$, then the random $d$-regular graph $G(n,d)$ can be ``sandwiched'' between $G(n,p_*)$ and $G(n,p^*)$ where $p_*$ and $p^*$ are both asymptotically equal to $d/n$. This famous conjecture was previously proved for all $d\gg (n\log n)^{3/4}$. In this paper, we confirm the conjecture when $d \gg \log^4 n$. We also extend this result to near-regular degree sequences.