# Fitting data to a log-normal distribution [duplicate]

For a simulation study I've been trying to find an appropriate distribution for job handling times in R. I have a very large dataset of 77010 records (handling time in seconds).

I've been exploring several distributions (lognormal, exponential, Gamma, Weibull, Burr, Pareto and more) and want to confirm which distribution best represents my data. I found that from all fitted distributions (using fitdist) the log normal distribution best represents my data (see picture below).

This plot makes me think that the lognormal distribution is a good representative of my data compared to the other distributions (also when comparing BIC, AIC & MLE). But, when I take the logs of my dataset and test for normality using ad.test (i.e., the Anderson Darling test), the null hypothesis is rejected. Likewise, when I perform the Kolmogorov-smirnov test (ks.test, note that ties are present here), the test is also rejected. I suspect my dataset is too large to obtain meaningful results from these tests.

How do I proceed in finding substantial evidence that this distribution fits my data?