  We study $C^{d,r}$-H\'enon-like families $(f_{a\, b})_{a\, b}$ with two
parameters $(a,b)\in \mathbb R^2$. We show the existence of an open set of
parameters $(a,b)\in \mathcal D$, so that a renormalization chart conjugates an
iterate of $f_{a\, b}$ to a perturbation of $(x,y)\mapsto ((x^2+c_1)^2+c_2,0)$.
We prove that the map $(a,b)\in \mathcal D\mapsto (c_1,c_2)$ is a
$C^d$-diffeomorphism; as first numerically conjectured by Milnor in 1992.
  Furthermore, we show the existence of an open set of parameters $(a,b)$ so
that $f_{a\, b}$ displays exactly two different renormalized H\'enon-like maps
whose basins union attracts Lebesgue a.e. point with bounded forward orbit. A
great freedom in the choice of the renormalized parameters enables us to deduce
in particular the existence of a (unperturbed) H\'enon map with exactly $2$
attracting cycles (an answer to a Question by Lyubich).
  The proof is based on a generalization of puzzle pieces for H\'enon-like
maps, and on a generalization of both the affine-like formalism of Palis-Yoccoz
and the cross map of Shilnikov. The distortion bounds enable us to define (for
the first time) $C^{r}$ and $C^{d,r}$-renormalizations and
multi-renormalizations with bounds on all the derivatives.
