We show that an a effect is driven by the cosmic-ray (CR) Bell instability exciting left-right asymmetric turbulence. Alfven waves of a preferred polarization have maximally helical motion, because the transverse motion of each mode is parallel to its curl. We show how large-scale Alfven modes, when rendered unstable by CR streaming, can create new net flux over any finite region, in the direction of the original large-scale field. We perform direct numerical simulations (DNSs) of a magnetohydrodynamic fluid with a forced CR current and use the test-field method to determine the alpha effect and the turbulent magnetic diffusivity. As follows from DNS, the dynamics of the instability has the following stages: (1) in the early stage, the small-scale Bell instability that results in the production of small-scale turbulence is excited; (2) in the intermediate stage, there is formation of larger-scale magnetic structures; (3) finally, quasi-stationary large-scale turbulence is formed at a growth rate that is comparable to that expected from the dynamo instability, but its amplitude over much longer timescales remains unclear. The results of DNS are in good agreement with the theoretical estimates. It is suggested that this dynamo is what gives weakly magnetized relativistic shocks such as those from gamma-ray bursts (GRBs) a macroscopic correlation length. It may also be important for large-scale magnetic field amplification associated with CR production and diffusive shock acceleration in supernova remnants (SNRs) and blast waves from GRBs. Magnetic field amplification by Bell turbulence in SNRs is found to be significant, but it is limited owing to the finite time available to the super-Alfvenicly expanding remnant. The effectiveness of the mechanisms is shown to be dependent on the shock velocity. Limits on magnetic field growth in longer-lived systems, such as the Galaxy and unconfined intergalactic CRs, are also discussed.