By Hironaka’s version of the Chow theorem (in turn a consequence of the flattening theorem), see [Hir75, Corollary 2], there exists a complex manifold ${\mathcal{X}}^{\prime\prime}$ and a projective bimeromorphic morphism ${\mathcal{X}}^{\prime\prime}\to{\mathcal{X}}$ such that ${\mathcal{X}}^{\prime\prime}$ dominates ${\mathcal{X}}^{\prime}$. Since ${\mathcal{X}}^{\prime}\to{\mathcal{X}}$ is an isomorphism above $X$, the construction in [Hir75] further guarantees that ${\mathcal{X}}^{\prime\prime}\to{\mathcal{X}}$ is an isomorphism above $X$. Indeed, the proof proceeds by blowing up well-chosen smooth centers contained in the non-flat locus of ${\mathcal{X}}^{\prime}\to{\mathcal{X}}$, see Définition 4.4.3 (2) in loc. cit.

We may therefore assume that ${\mathcal{X}}^{\prime}\to{\mathcal{X}}$ itself is projective, and more precisely the blowup of an ideal $I$ cosupported on ${\mathcal{X}}_{0}$. By the principalization theorem for ideals, there exists a projective bimeromorphic morphism ${\mathcal{X}}^{\prime\prime}\to{\mathcal{X}}$ that is a composition of simple blowups, such that the pullback of $I$ to ${\mathcal{X}}^{\prime\prime}$ is a principal ideal, see [Kol07, Theorem 3.45] or [Wło09, Theorem 2.0.3]. In particular, ${\mathcal{X}}^{\prime\prime}$ dominates ${\mathcal{X}}^{\prime}$. ∎