In a paper on Hadwiger's conjecture, https://web.math.princeton.edu/~pds/papers/hadwiger/paper.pdf, Seymour explains various results on excluding the complete graph as a minor.

In particular, there is a nice bound on the number of edges, due to Mader, as follows:

If $t\leq 7$ and $n\geq t-2$ then every $K_t$-minor free graph $G$ on $n$ vertices has $$ |E(G)|\leq (t-2)n-\frac{(t-1)(t-2)}{2}.$$

(The obvious application to Hadwiger, as Seymour describes, is that this bound controls the average degree and hence gives low minimum degree to which you can use induction to get a colouring result.)

Sadly the pattern fails for $t\geq 8$. Jorgensen and then Song and Thomas describe the exceptions explicitly for $t=8$ and $t=9$ respectively but as far as I can see no larger $t$ is understood. To be more precise Jorgensen's result says that if there is no $K_8$-minor and the displayed inequality fails then the exact number of edges is known and $G$ can be built from $K_{2,2,2,2,2}$ in a simple way (by "pure 5-sums").

However, in general the average degree in a $K_t$-minor free graph can be large - Kostochka, Thomason, ... - so the "exceptions" will get really bad for large $t$.

I'm interested in this from a slightly different perspective. Are explicit descriptions known (or potentially tractable) of $K_t$-minor free graphs for general $t$ provided that $n$ is quite small relative to $t$?

So a desired result would be something like:

For any natural number $t$ and any $t-2\leq n \leq 2t$, every $K_t$-minor free graph $G$ on $n$ vertices has $$ |E(G)|\leq (t-2)n-\frac{(t-1)(t-2)}{2},$$ unless $G$ is ... some exceptions like those in Jorgensen/Song and Thomas ...

If this is somehow easy then replace the vertex bound of $2t$ with something bigger.