The seminal 1975 work of Brascamp-Lieb-Lebowitz initiated the rigorous study of Ginzberg-Landau random surface models. It was conjectured therein that fluctuations are localized on $\mathbb Z^d$ when $d\geq 3$ for very general potentials, matching the behavior of the Gaussian free field. We confirm this behavior for all even potentials $U:\mathbb R\to\mathbb R$ satisfying $U'(x)\geq \min(\varepsilon x,\frac{1+\varepsilon}{x})$ on $x\in \mathbb R^+$. Given correspondingly stronger growth conditions on $U$, we show power or stretched exponential tail bounds on all transient graphs, which determine the maximum field value up to constants in many cases. Further extensions include non-wired boundary conditions and iterated Laplacian analogs such as the membrane model. Our main tool is an FKG-based generalization of the Gaussian correlation inequality, which is used to dominate the finite-volume Gibbs measures by mixtures of centered Gaussian fields.