Why do confidence intervals width increase when sample size increases "a bit"? I'll try to explain better. I sampled a LogNormal random variate and I extracted 12001 and 12002 samples (with same initial seed for Random Number Generator). In the first case the half width of confidence interval (for mean) is 0.36633536625893326, while in the second case the width is 0.36636953704150527. Comparing two numbers 0.36633536625893326 >= 0.36636953704150527 is false.
So I don't understand why width is increased. I know that when sample size increases confidence intervals decrease. Maybe did I make some mistakes? Or is this a normal behavior?

A standard confidence interval for a mean is calculated using $s/\sqrt n$. If you increase $n$ but also increase the sample standard deviation $s$ by enough to offset the larger sample size, then your $s/\sqrt n$ increases, widening your confidence interval.
$$\text{Standard Confidence Interval for a Mean}\\
\bigg(
\bar x - t_{critical}\dfrac{s}{\sqrt n},
\bar x + t_{critical}\dfrac{s}
{\sqrt n}
\bigg)
$$
