Every (real or complex) vector space $E$ can be endowed with its finest locally convex topology $\tau_{flc}$ where every seminorm is continuous and (equivalently) every absolutely convex set is a $0$-neighborhood.
This topology can be described as the locally convex inductive limit of all finite dimensional subspaces (with their unique Hausdorff locally convex topologies). Since bornologicity is stable with respect to inductive limits the finest locally convex topology is thus bornological (also ultrabornological). Using e.g. a Hamel basis of $E$ you find linear functionals which are discontinuous with respect to the Hilbert space topology. As they are clearly continuous with respect to $\tau_{flc}$ we get that the latter is strictly finer than the Hilbert topology.
The identity $(E,\|\cdot\|) \to (E,\tau_{flc})$ has closed graph (since it is continuous in the other direction) but is not continuous. This shows that being webbed cannot be dropped from the closed graph theorem and that $(E,\tau_{lfc})$ is not webbed.