Let $D$ be an effective Cartier divisor on a normal noetherian scheme $X$. Its irreducible components are codimension $1$ subschemes, i.e. Weil divisors, of $X$ but not necessarily Cartier divisors. I would like to construct a birational morphism $f:X' \to X$ such that the irreducible components of the pullback $f^*D$ are Cartier divisors. If $f$ were only an alteration, that would also be fine.
If we assume resolution of singularities, this is possible: Just choose $f$ such that $X'$ is regular. Then every Weil divisor is Cartier and we are done. Is there an argument avoiding resolution of singularities (and the theory of alterations)?
