We consider immersed hypersurfaces in euclidean $R^{n+1}$ which are stable with respect to an elliptic parametric functional with integrand $F=F(N)$ depending on normal directions only. We prove an integral curvature estimate provided that $F$ is sufficiently close to the area integrand, extending the classical curvature estimate of Schoen, Simon and Yau for stable minimal hypersurfaces in $R^{n+1}$. As a crucial point of our analysis we derive a generalized Simons inequality for the laplacian of the length of a weighted second fundamental form with respect to an abstract metric associated with $F$. Using Moser's iteration technique we finally prove a pointwise curvature estimate for $n leq 5$. As an application we obtain a new Bernstein result for complete stable hypersurfaces of dimension $n leq 5$.