Note that $\min(w,L) = \frac{1}{2} (w+L) + - \frac{1}{2} |w-L|$, so the main issue is to establish that the map $w \mapsto |w|$ is bounded on $H^{1,2}$. But this follows from the diamagnetic inequality $|\nabla |w|| \leq |\nabla w|$ (in the sense of distributions), which is obvious formally, but can be established rigorously by approximating the absolute value $|x|$ by the smoothed variant $(|x|^2+\varepsilon^2)^{1/2}$ and then letting $\varepsilon$ go to zero. (The diamagnetic inequality can be found for instance in the text of Lieb and Loss; it is more commonly applied in the context of covariant differentiation, but already has usefully non-trivial content for ordinary differentiation.)
More generally, composition with Lipschitz functions will preserve all $W^{s,p}(\Omega)$ spaces for $0 \leq s \leq 1$ and $1 < p < \infty$ by the chain rule (if $s=1$) or fractional chain rule (if $s<1$), using regularisation arguments as necessary to make the argument rigorous. (See for instance Taylor's book "Tools for PDE" for this sort of thing.)
Note that $\min(w,L) = \frac{1}{2} (w+L) + \frac{1}{2} |w-L|$, so the main issue is to establish that the map $w \mapsto |w|$ is bounded on $H^{1,2}$. But this follows from the diamagnetic inequality $|\nabla |w|| \leq |\nabla w|$ (in the sense of distributions), which is obvious formally, but can be established rigorously by approximating the absolute value $|x|$ by the smoothed variant $(|x|^2+\varepsilon^2)^{1/2}$ and then letting $\varepsilon$ go to zero. (The diamagnetic inequality can be found for instance in the text of Lieb and Loss; it is more commonly applied in the context of covariant differentiation, but already has usefully non-trivial content for ordinary differentiation.)
More generally, composition with Lipschitz functions will preserve all $W^{s,p}(\Omega)$ spaces for $0 \leq s \leq 1$ and $1 < p < \infty$ by the chain rule (if $s=1$) or fractional chain rule (if $s<1$), using regularisation arguments as necessary to make the argument rigorous. (See for instance Taylor's book "Tools for PDE" for this sort of thing.)