Let $X$ be a complex variety whose singular locus is a smooth variety $Z$.  
Let $f:Y\rightarrow X$ be a small resolution of $X$ such that $f^{-1}(z)$ is smooth for any $z\in Z$ and $dim(f^{-1}(z))$ is constant on $Z$.  
Is it true that the blow-up $Bl_Z(X)$ of $X$ along $Z$ is normal (then it would be even smooth by Zariski main theorem)?