On page 7 of:http://arxiv.org/pdf/0812.1043.pdf Guth and Katz apply lemma 4.1 to prove that $|J_r|\geq 999/1000 |J|$. This application should be fairly trivial, but with a lot of staring I can't finish it. I'll give my best effort and hope to indicate where I get stuck.
We need to set $\mu$ as the minimum degree of the set $Y'=L_R$, which should be $\mu = KN^{1/2}/1000.$ We then need $\rho$ as in $|E|<\rho |Y|=\rho|L|$. We know that there are at least 3|J| edges and $|J|>KN^{3/2}$ by assumption, so we can set $\rho=3KN^{1/2}$
Applying the theorem gives $$|J_R|=|E'|<(3KN^{1/2}-\frac{KN^{1/2}}{1000})N$$ $$=(3-1/1000)KN^{3/2} $$
This is very close to the answer, as if $KN^{3/2}>|J|$ we would be done, but the assumption is the opposite.