Summary: Let \(d\geq 1\) be fixed. Let \(F\) be a number field of degree \(d\), and let \(E/F\) be an elliptic curve. Let \(E(F)_{\mathrm{tors}}\) be the torsion subgroup of \(E(F)\). In 1996, L. Merel [Invent. Math. 124, No. 1–3, 437–449 (1996; Zbl 0936.11037)] proved the uniform boundedness conjecture, i.e., there is a constant \(B(d)\), which depends on \(d\) but not on the chosen field \(F\) or on the curve \(E/F\), such that the size of \(E(F)_{\mathrm{tors}}\) is bounded by \(B(d)\). Moreover, Merel gave a bound (exponential in \(d\)) for the largest prime that may be a divisor of the order of \(E(F)_{\mathrm{tors}}\). In 1996, P. Parent [J. Théor. Nombres Bordx. 15, No. 3, 831–838 (2003; Zbl 1072.11037)] proved a bound (also exponential in \(d\)) for the largest \(p\)-power order of a torsion point that may appear in \(E(F)_{\mathrm{tors}}\). It has been conjectured, however, that there is a bound for the size of \(E(F)_{\mathrm{tors}}\) that is polynomial in \(d\). In this article we show that if \(E/F\) has potential supersingular reduction at a prime ideal above \(p\), then there is a linear bound for the largest \(p\)-power order of a torsion point defined over \(F\), which in fact is linear in the ramification index of the prime of supersingular reduction.