RCT and data normality are not related or contradictory concepts.
Randomized controlled trial simply implies that the treatment was allocated in a random manner and the active treatment is controlled with sham or equivalents.
In RCT, data can follow or not normal distribution and this will guide the choice of your statistical methods.
You can check for data normality using several tests such as the Kolmogorov-Smirnov test.
If the data follow normal distribution, you can apply parametric tests (ex: t test or ANOVA for group comparison, and Pearson correlation coefficient for correlation analysis).
If the data did not follow normal distribution, the choice would be non-parametric tests such as Mann-Whitney for group comparison or Spearman correlation coefficient for correlation analysis.