  A family $\mathcal{F}$ of elliptic curves defined over number fields is said
to be typically bounded in torsion if the torsion subgroups $E(F)[$tors$]$ of
those elliptic curves $E_{/F}\in \mathcal{F}$ can be made uniformly bounded
after removing from $\mathcal{F}$ those whose number field degrees lie in a
subset of $\mathbb{Z}^+$ with arbitrarily small upper density. For every number
field $F$, we prove unconditionally that the family $\mathcal{E}_F$ of elliptic
curves defined over number fields and with $F$-rational $j$-invariant is
typically bounded in torsion. For any integer $d\in\mathbb{Z}^+$, we also
strengthen a result on typically bounding torsion for the family
$\mathcal{E}_d$ of elliptic curves defined over number fields and with degree
$d$ $j$-invariant.
