On the coloring number of small graphs with small cliques
Uniformly random $k^2$-vertex graphs have clique size $O(\log k)$, well under $k$, and independent set size also $O(\log k)$, implying that their chromatic number is $\Omega(k^2/\log k)$.

As for $k^2$-vertex triangle-free graphs, their chromatic number can be $\Theta(k/\sqrt{\log k})$ (and not higher); see Kim, Jeong Han (1995),
"The Ramsey number $R(3,t)$ has order of magnitude $t^2/\log t$", Random Structures &amp; Algorithms 7 (3): 173–207, doi:10.1002/rsa.3240070302

If by "size" you mean edges rather than vertices, then the maximum chromatic number of triangle-free $k^2$-edge graphs is $\Theta((k/\log k)^{2/3})$. See Nilli, A. (2000), "Triangle-free graphs with large chromatic numbers", Discrete Mathematics 211(1–3): 261–262, doi:10.1016/S0012-365X(99)00109-0.