Let $n:\widetilde X\rightarrow X$ be the normalization of a complex (quasi-projective) variety $X$. Assume $\widetilde X$ is smooth, that $n$ is an isomorphism outside a smooth connected subvariety $Y\subset \widetilde X$ of codimension $>1$ and that $n_{|Y}:Y\rightarrow n(Y)$ has degree $2$ (the quotient of $Y$ by an involution).
Are the blow-ups $Bl_Y(\widetilde X)$ and $Bl_{n(Y)}(X)$ necessarily isomorphic?