Let $f(n)\to \infty$ possibly very slowly. Is there a graph of infinite chromatic number such that every finite subgraph on $n$ vertices can be made bipartite by deleting at most $f(n)$ edges?

The set of minimum edge distances to bipartite for subgraphs of size n is bounded above. A graph on n vertices has at most n choose 2 edges, and deleting all of them makes the graph bipartite, providing a straightforward upper bound.