weights in low density codes
Your question is somewhat ill-posed: given $n$ and $k$, it is easy to construct a Tanner graph with a 4-cycle and column weight at most 2.

Instead, you can ask the question of what is the maximum column weight of a Tanner graph, given that is has no 4-cycle. Assuming uniform column weights, then the maximum column weight is $O(\sqrt{k})$. This is known as the Kővári–Sós–Turán theorem (see e.g. here), which states that the number of edges in an $(n,k)$-bipartite graph without 4-cycle is at most
$$
(n-1)\sqrt{k} + k.
$$
If the Tanner graph has uniform column weights $d$, then it has $nd$ edges, which gives the bound on $d$.

This is tight up to constants for the case where $k = n$ (see e.g. here).