Fluid elements deform in turbulence by stretching and folding. In this Letter, by projecting the material deformation tensor onto the largest stretching direction, we depict the dynamics of folding through the evolution of the material curvature. Results from direct numerical simulation (DNS) show that the curvature growth exhibits two regimes: first, a linear stage dominated by folding fluid elements through a persistent velocity Hessian that then transition to an exponential-growth stage driven by the stretching of already strongly bent fluid elements. This transition leads to strong curvature intermittency at later stages, which can be explained by a proposed curvature-evolution model. The link between velocity Hessian to folding provides a new way to understand the crucial steps in energy cascade and mixing in turbulence beyond the classical linear description of stretching dynamics.