Bootstrapping does not increase the sample size. You usually use bootstrapping (drawing from the data with replacement) to create multiple bootstrap samples of the same size as your original data in order to calculate e.g. a standard error or confidence interval for something you calculate on your original data. E.g. you have $n$ patients in two groups (of size $n_1$ and $n_2$), and want a confidence interval for the difference between the group means (that you simply calculate on the original data), in which case you could draw repeatedly (e.g. a few thousand times) $n$ patients from your data, calculate the difference between the groups and take the 2.5th and 97.5th percentile of the calculated differences as a (quantile bootstrap) confidence interval.
Additionally, more data that does not quite follow an assumed distribution does not somehow make the data follow the assumed distribution.
Also note: small deviations from normal residual are not necessarily a problem. There is a substantial literature on how linear regression is reasonably robust to small to moderate sized deviations for the kind of sample sizes you are talking about.