The fragmentation of small, brittle, flexible, inextensible fibers is investigated in a fully-developed, homogeneous, isotropic turbulent flow. Such small fibers spend most of their time fully stretched and their dynamics follows that of stiff rods. They can then break through tensile failure, i.e. when the tension is higher than a given threshold. Fibers bend when experiencing a strong compression. During these rare and intermittent buckling events, they can break under flexural failure, i.e. when the curvature exceeds a threshold. Fine-scale massive simulations of both the fluid flow and the fiber dynamics are performed to provide statistics on these two fragmentation processes. This gives ingredients for the development of accurate macroscopic models, namely the fragmentation rate and daughter-size distributions, which can be used to predict the time evolution of the fiber size distribution. Evidence is provided for the generic nature of turbulent fragmentation and of the resulting population dynamics. It is indeed shown that the statistics of breakup is fully determined by the probability distribution of Lagrangian fluid velocity gradients. This approach singles out that the only relevant dimensionless parameter is a local flexibility which balances flow stretching to the fiber elastic forces.