  The present paper studies a quantitative version of the transversality
theorem. More precisely, given a continuous function $g\in
\mathcal{C}([0,1]^d,\mathbb{R}^m)$ and a global smooth manifold $W\subset
\mathbb{R}^m$ of dimension $p$, we establish a quantitative estimate on the
$(d+p-m)$-dimensional Hausdorff measure of the set
$\mathcal{Z}_{W}^{g}=\left\{x\in [0,1]^d: g(x)\in W\right\}$. The obtained
result is applied to quantify the total number of shock curves in weak entropy
solutions to scalar conservation laws with uniformly convex fluxes in one space
dimension.
