We here examine the structure of turbulence in the case of a complex fluid made up of water and surfactants. This fluid has the particular property of shear thickening when driven at shear rates above a certain threshold. Through a study of the spectral properties and the structure function scalings, important differences arise with respect to the reference case, i.e., water. The surfactant solution shows strong intermittency at small scales. The large scales are, on the other hand, free of intermittency. While this transition is observed in the structure function scalings, no sign of this transition is seen in the power spectrum of velocity fluctuations which shows a single scaling range. The strongly intermittent small-scale region, despite the scaling of the power spectrum, exhibits properties reminiscent of the near-dissipative range.