  For $n\in \mathbb{N}$, let $Y_n$ denote the linear span of the first $n+1$
levels of the Haar system in a Haar system Hardy space $Y$ (this class contains
all separable rearrangement-invariant function spaces and also related spaces
such as dyadic $H^1$). Let $I_{Y_n}$ denote the identity operator on $Y_n$. We
prove the following quantitative factorization result: Fix
$\Gamma,\delta,\varepsilon > 0$, and let $n,N \in \mathbb{N}$ be chosen such
that $N \ge Cn^2$, where $C = C(\Gamma,\delta,\varepsilon) > 0$ (this amounts
to a quasi-polynomial dependence between $\dim Y_N$ and $\dim Y_n$). Then for
every linear operator $T\colon Y_N\to Y_N$ with $\|T\|\le \Gamma$, there exist
operators $A,B$ with $\|A\|\|B\|\le 2(1+\varepsilon)$ such that either $I_{Y_n}
= ATB$ or $I_{Y_n} = A(I_{Y_N} - T)B$. Moreover, if $T$ has $\delta$-large
positive diagonal with respect to the Haar system, then we have $I_{Y_n} = ATB$
for some $A,B$ with $\|A\|\|B\|\le (1+\varepsilon)/\delta$. If the Haar system
is unconditional in $Y$, then an inequality of the form $N \ge Cn$ is
sufficient for the above statements to hold (hence, $\dim Y_N$ depends
polynomially on $\dim Y_n$). Finally, we prove an analogous result in the case
where $T$ has large but not necessarily positive diagonal entries.
