The hurdle (and zero-inflation) count regression models typically use a log-link for the count component of the model. So you already have a log-linear relationship between the expectation of the (latent) count component and the regressors already. Hence, further transformations are usually not necessary/applicable.
Furthermore, a sqrt-transformation of the response does not yield count data anymore, hence this does not make much sense for count data.
I see three possible actions for you:
- Do not look at just the residuals but also compare observed and expected frequencies for 0, 1, 2, ... counts. Often the fitted distribution is quite satisfactory even if the residuals are large - especially when the overdispersion is substantial (i.e., low theta in the negbin distribution).
- You have to consider either a different count distribution and/or improve the specification of the regressors in your model to obtain a better fit.
- If your counts are large enough, it may be possible to use a continuous rather than a discrete distribution. Thus, instead of the zero-truncated negative binomial distribution, you could use a zero-truncated normal distribution - either for the original response or after a possible sqrt-transformation etc.
Which of these is most appropriate depends on your data, sample size, and the kind of predictions you want to make.