  Watkins's conjecture suggests that for an elliptic curve $E/\mathbb{Q}$, the
rank of the group $E(\mathbb{Q})$ of rational points is bounded above by $\nu_2
(m_E)$, where $m_E$ is the modular degree associated with $E$. It is known that
Watkins's conjecture holds on average. This article investigates the conjecture
over certain thin families of elliptic curves. For example, for prime $\ell$,
we quantify the elliptic curves featuring a rational $\ell$-torsion that
satisfies Watkins's conjecture. Additionally, the study extends to a broader
context, investigating the inequality $\mathrm{rank}(E(\mathbb{Q}))+M\leq
\nu_2(m_E)$ for any positive integer $M$.
