We investigate the impact of a stochastic background of primordial magnetic fields (PMF) generated before recombination on the ionization history of the Universe and on the cosmic microwave background (CMB) radiation. Pre-recombination PMFs are dissipated during recombination and reionization via decaying MHD turbulence and ambipolar diffusion. This modifies the local matter and electron temperatures and thus affects the ionization history and Thomson visibility function. We use this effect to constrain PMFs described by a spectrum of power-law type extending our previous study (based on a scale-invariant spectrum) to arbitrary spectral index, assuming that the fields are already present at the onset of recombination. We improve previous analyses by solving several numerical issues which appeared for positively tilted PMFs indices. We derive upper bounds on the integrated amplitude of PMFs due to the separate effect of ambipolar diffusion and MHD decaying turbulence and their combination. We show that ambipolar diffusion is relevant for nB > 0 whereas for nB < 0 MHD turbulence is more important. The bound marginalized over the spectral index on the integrated amplitude of PMFs with a sharp cut-off is \sqrt{< B^2 \rangle }\lt 0.83 nG. We discuss the quantitative relevance of the assumptions on the damping mechanism and the comparison with previous bounds.