Let $A=(a_{ij})$ be an $n \times n$ row-stochastic matrix, and its column sums are of order $m$, that is, $max_j\{\sum_{i=1}^n a_{ij}\}=O(m)$, where $m$ could depend on $n$.
Is it true that the column sums of $A^2$ will be of the same order $m$?
It seems to me so, but I couldn't prove it.
I don't have a complete answer, but maybe the following consideration may help. Clearly, $A^2$ is row-stochastic, and suppose the statement is true that the column sums are of order $m$ (i.e., you assertion is correct). Then, you can show by induction that the column sums of $A^\ell$ are of order $m$, for every $\ell$. If $A$ is irreducible and aperiodic, and if $\pi^T=\pi^TA$, then $A^\ell$ converges to $1\pi^T$. Hence,
$$ \max_j \sum_{i=1}^n a^\ell_{ij} \to n\cdot \max_j \pi_j. $$
Right?