  Let $\mathbb{F}_q$ be a finite field of odd characteristic $p$. We exhibit
elliptic curves over the rational function field $K = \mathbb{F}_q(t)$ whose
Tate-Shafarevich groups are large. More precisely, we consider certain infinite
sequences of explicit elliptic curves $E$, for which we prove that their
Tate-Shafarevich group $\mathrm{III}(E)$ is finite and satisfies
$|\mathrm{III}(E)| = H(E)^{1+o(1)}$ as $H(E)\to\infty$, where $H(E)$ denotes
the exponential differential height of $E$. The elliptic curves in these
sequences are pairwise neither isogenous nor geometrically isomorphic. We
further show that the $p$-primary part of their Tate-Shafarevich group is
trivial. The proof involves explicitly computing the $L$-functions of these
elliptic curves, proving the BSD conjecture for them, and obtaining estimates
on the size of the central value of their $L$-function.
