If $E$ is an elliptic curve over $K$, is there any effective estimate for the discriminant of the extension $L/K$ for which the $p$-part of the Selmer or Tate-Shafarevich groups become large?

I will add this edit to make the question more precise:

while going up any tower of number fields, do we fully understand the growth of $Sel_p$?

Given $E/K$ is it possible/ is it known that there will always exist an extension $M/K$ such that $Sel_p(E/M)$ is large (say $> p^m$ for any integer $m$, and $M$ depends on $m$ somehow)? In that case, what is the effective bound on $M$?

Kloosterman-Schaeffer (2003) prove that for $p>3$, there are infinitely many elliptic curves defined over $K$ with Selmer group large when $[K:\mathbb{Q}]\leq g+1$ ($g=$ genus of $X_0(p)$). Are there known effective bounds? If one can't give an effective bound in terms of the degree, is it possible to give one in terms of the discriminant?