Type Ia supernovae (SNe) are thought to originate from the thermonuclear explosions of carbon-oxygen (CO) white dwarfs (WDs). The proposed progenitors of standard type Ia SNe have been studied for decades and can be, generally, divided into explosions of CO WDs accreting material from stellar non-degenerate companions (single-degenerate; SD models), and those arising from the explosive interaction of two CO WDs (double-degenerate; DD models). However, current models for the progenitors of such SNe fail to reproduce the diverse properties of the observed explosions, nor do they explain the inferred rates and the characteristics of the observed populations of type Ia SNe and their expected progenitors. Here we show that the little-studied mergers of CO-WDs with hybrid Helium-CO (He-CO) WDs can provide for a significant fraction of the normal type Ia SNe. Here we use detailed thermonuclear-hydrodynamical and radiative-transfer models to show that a wide range of mergers of CO WDs with hybrid He-CO WDs can give rise to normal type Ia SNe. We find that such He-enriched mergers give rise to explosions for which the synthetic light-curves and spectra resemble those of observed type Ia SNe, and in particular, they can produce a wide range of peak-luminosities, MB(MR)~ 18.4 to 19.2 (~ 18.5 to 19:45), consistent with those observed for normal type Ia SNe. Moreover, our population synthesis models show that, together with the contribution from mergers of massive double CO-WDs (producing the more luminous SNe), they can potentially reproduce the full range of type Ia SNe, their rate and delay-time distribution.