We study magnetogenesis in axionlike inflation driven by a pseudoscalar field phi coupled axially to the electromagnetic (EM) field (beta/M-p)phi F-mu nu(F) over tilde mu nu with dimensionless coupling constant beta. A set of equations for the inflaton field, scale factor, and expectation values of quadratic functions of the EM field is derived. These equations take into account the Schwinger effect and the backreaction of generated EM fields on the Universe expansion. It is found that the backreaction becomes important when the EM energy density reaches the value rho(EM) similar to (root 2 epsilon/beta)rho(inf) (epsilon is the slow-roll parameter and rho(inf) is the energy density of the inflaton) slowing down the inflaton rolling and terminating magnetogenesis. The Schwinger effect becomes relevant when the electric energy density exceeds the value rho(E) similar to alpha(-3)(EM)(rho(2)(tot)/M-p(4)), where rho(tot) = 3H(2)M(p)(2) is the total energy density and alpha(EM) is the EM coupling constant. For large beta, produced charged particles could constitute a significant part of the Universe energy density even before the preheating stage. Numerically studying magnetogenesis in the alpha-attractor model of inflation, we find that it is possible to generate helical magnetic fields with the maximal strength 10(-15) G, however, only with the correlation length of order 1 pc at present.