This paper proposes a data-driven control design method for nonlinear systems that builds upon the Koopman operator framework. In particular, the Koopman operator is used to lift the nonlinear dynamics to a higher-dimensional space where the so-called observables evolve linearly. First, an approximate linear time-invariant (LTI) lifted representation of the nonlinear system is obtained. To take into account the residual error, an approximation of the $\ell_2$-gain of the error system is computed from data. Based on the obtained model and the $\ell_2$-gain bound, a dynamic feedback controller providing robust performance guarantees is synthesized. The controller synthesis method only depends on the frequency response of the LTI approximation; thus, is independent of the lifting dimension. Next, to further reduce the $\ell_2$-gain of the error system, a linear parameter-varying (LPV) lifted model is considered. A control design method based on the robust control of the LTI part of dynamics and compensation of the parameter-varying dynamics is proposed. It is shown that the presented control strategy guarantees internal stability of the closed-loop system under the assumption that the parameter-varying dynamics are open-loop BIBO stable while also delivering robust performance guarantees for certain input-output channels through which the parameter-varying dynamics are fully cancelled.