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We propose and validate a novel experimental technique to measure two-point statistics of turbulent flows. It consists in spreading rigid fibers in the flow and tracking their position and orientation in time and therefore been named ``Fiber Tracking Velocimetry'' (FTV). By choosing different fiber lengths, i.e. within the inertial or dissipative range of scales, the statistics of turbulence fluctuations at the selected lengthscale can be probed accurately by simply measuring the fiber velocity at its two ends, and projecting it along the transverse-to-the-fiber direction. By means of fully-resolved direct numerical simulations and experiments, we show that these fiber-based transverse velocity increments are statistically equivalent to the (unperturbed) flow transverse velocity increments. Moreover, we show that the turbulent energy dissipation rate can be accurately measured exploiting sufficiently short fibers. The technique has been tested against standard Particle Tracking Velocimetry (PTV) of flow tracers with excellent agreement. Our technique overcomes the well-known problem of PTV to probe two-point statistics reliably because of the fast relative diffusion in turbulence that prevents the mutual distance between particles to remain constant at the lengthscale of interest. This problem, making it difficult to obtain converged statistics for a fixed separation distance, is even more dramatic for natural flows in open domains. A prominent example are oceanic currents, where drifters (i.e.~the tracer-particle counterpart used in field measurements) disperse quickly, but at the same time their number has to be limited to save costs. Inspired by our laboratory experiments, we propose pairs of connected drifters as a viable option to solve the issue.