Support of homology in quasi-projective varieties.

Question

Given a quasi-projective complex variety $X$ and a positive integer $i<\text{dim}(X)-1$. Consider the homology group $H_i(X(\mathbb{C}))$. Is it possible to find a subvariety of codimension at least $1$ inside $X$, denoted by $Y$ such that $H_i(Y)\cong H_i(X)$? Note that this is true if replace quasi-projective by projective (Lefschetz hyperplane), or the homology with Borel-Moore homology.