Comparisons to Measles are somewhat misleading. Omicron spreads quite rapidly. Doubling times of ~2-3 days. But its fast growth can be a combination of a high R0 or a short generation time (time between infections), or both.
Early data seems to suggest it is both. Speed of spread in an epidemic is driven both by R0 and the generation time. Measles has a very high R0 (~18), but a long generation time (~2 weeks). We don't yet have precise estimates for Omicron's R0. Generation time is usually inferred from serial interval (time between symptomatic cases). The wild type SARS-COV-2's R0 was estimated at ~2.5 and its serial time was estimated around ~5 days. So even though it had a much lower R0 than Measles, the shorter generation time meant it could spread very rapidly.
R0 can affect the size of the epidemic as the herd immunity threshold is a function of R0. For example the rough herd immunity threshold of Measles is ~94%. For Influenza with R0 of ~1.3-2, it is more like ~30-50%.
Relative Rt estimates will be affected by immune evasion. We know Omicron has significant immune evasion, so it has a much larger pool of susceptible people to infect than Delta and so can get a transmission advantage purely from that. Likewise Rt estimates are sensitive to assumptions of the generation time, as a shorter generation time can be responsible for the apparent high number of new cases.