The edge-density for K2, t minors

Let H be a graph. If G is an n-vertex simple graph that does not contain H as a minor, what is the maximum number of edges that G can have? This is at most linear in n, but the exact expression is known only for very few graphs H. For instance, when H is a complete graph K"t, the ''natural'' conjecture, (t-2)n-12(t-1)@?(t-2), is true only for t= =2) the truth for infinitely many n.