In a typical X-ray spectrum we can observe that characteristic X-rays occur in noticeably larger numbers compared to other energies.

Characteristic emission is a separate phenomenon from bremsstrahlung. Characteristic emission is unique to a distinct set of energies, but bremsstrahlung can produce X-rays of any energy, including the characteristic energy. So the effect of characteristic emission is "stacked" on top of the bremsstrahlung effect, and this is why we can observe spikes on the spectrum. Does this make sense?