We present simulation results of vortex-induced vibrations of an infinitely long flexible cylinder at Reynolds number Re = 1000, corresponding to a ‘young’ turbulent wake (i.e. exhibiting a small inertial subrange). The simulations are based on a new class of spectral methods suitable for unstructured and hybrid grids. To obtain different responses of the coupled flow–structure system we vary the structure's bending stiffness to model the behaviour of a vibrating inflexible (rigid) cylinder, a cable, and a beam. We have found that unlike the laminar flow previously studied, the amplitude of the cross-flow oscillation is about one diameter for the cable and the beam, close to experimental measurements, but is lower for the rigid cylinder. We have also found that for the latter case the flow response corresponds to parallel shedding, but for the beam and cable with free endpoints a mixed response consisting of oblique and parallel shedding is obtained, caused by the modulated travelling wave motion of the structure. This mixed shedding pattern which alternates periodically along the span can be directly related to periodic spatial variation of the lift force. In the case of structures with pinned endpoints a standing wave response is obtained for the cylinder; lace-like flow structures are observed similar to the ones seen in the laminar regime. Examination of the frequency spectra in the near wake shows that at Re = 1000 all cases follow a −5/3 law in the inertial range, which extends about half a decade in wavenumber. However, these spectra are different in all three cases both in low and high frequencies, with the exception of the beam and cable, for which the high-frequency portion is identical despite the differences in the displacement time history and the large-scale features of the corresponding flow.