We report an experimental study of the magnetic field B--> induced by a turbulent swirling flow of liquid sodium submitted to a transverse magnetic field B-->(0). We show that the induced field can behave nonlinearly as a function of the magnetic Reynolds number, R(m). At low R(m), the induced mean field along the axis of the flow, <Bx>, and the one parallel to B-->(0), <B(y)>, first behave like R(2)(m), whereas the third component, <B(z)>, is linear in R(m). The sign of <Bx> is determined by the flow helicity. At higher R(m), B--> strongly depends on the local geometry of the mean flow: <Bx> decreases to zero in the core of the swirling flow but remains finite outside. We compare the experimental results with the computed magnetic induction due to the mean flow alone.