Turán densities for matroid basis hypergraphs

Let $U$ be a uniform matroid. For all positive integers $n$ and $r$ with $n\ge r$, what is the maximum number of bases of an $n$-element, rank-$r$ matroid without $U$ as a minor? We show that this question arises by restricting the problem of determining the Turán number of a daisy hypergraph to the family of matroid basis hypergraphs. We then answer this question for several interesting choices of $U$.