The effect of Raman instabilities on the production of fast electrons in laser–plasma interaction has been investigated for laser intensities well above the electron trapping threshold. The results of one-dimensional particle-in-cell simulations show that in this regime the presence of Raman backscattering (RBS) hampers fast-electron production, and that its suppression increases the yield of high-energy electrons ($>$15 MeV). Such suppression has been realized either through deletion of all backscattered radiation from the simulations or through direct stimulation of Raman forward scattering (RFS). An increased high-energy electron yield has been observed for both methods. In addition, the influence of various laser and plasma parameters on the production of highly energetic electrons has been investigated. Specifically, the effects of plasma density ramps, skews in the temporal envelopes of the laser pulses, and laser frequency chirp (both pulse-length preserving and bandwidth preserving) have been examined. For each parameter, its influence on the yield of high-energy electrons can be explained from the way it affects the balance between RBS and RFS excitation in laser–plasma interaction.