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In this paper, we prove that integral $n$-varifolds $\mu$ in codimension 1 with $H_{\mu }\in L_{\mathrm{loc}}^p\left(\mu \right)$, $p\>n$, $p\ge 2$ have quadratic tilt-excess decay ${\mathrm{tiltex}}_{\mu }(x,\varrho ,T_x\mu )=O_x\left({\varrho }^2\right)$for $\mu$-almost all $x$, and a strong maximum principle which states that these varifolds cannot be touched by smooth manifolds whose mean curvature is given by the weak mean curvature $H_{\mu }$, unless the smooth manifold is locally contained in the support of $\mu$.