We explain how to use results from Iwasawa theory to obtain information about \(p\)-parts of Tate-Shafarevich groups of specific elliptic curves over \({\mathbb{Q}}\). Our method provides a practical way to compute \(\#\Sha(E/{\mathbb{Q}})(p)\) in many cases when traditional \(p\)-descent methods are completely impractical and also in situations where results of Kolyvagin do not apply, e.g., when the rank of the Mordell-Weil group is greater than 1. We apply our results along with a computer calculation to show that \(\Sha(E/{\mathbb{Q}})[p]=0\) for the 1,534,422 pairs \((E,p)\) consisting of a non-CM elliptic curve \(E\) over \({\mathbb{Q}}\) with conductor \({\leqslant}30,\!000\), rank \({\geqslant}2\), and good ordinary primes \(p\) with \(5 {\leqslant}p < 1000\) and surjective mod-\(p\) representation.