Random Graph is a good expander
Try proving directly that with high probability, no set of size at most $n/2$ has a small edge neighborhood. See for example Ellis's writeup for the complete details.

Another possibility is to try and use a more sophisticated large deviation bound to analyze $|E(S,S')|$. For example, there are versions of Chernoff's bound for negatively dependent random variables, and there is Azuma's bound, which is a martingale version of Chernoff's bound. The classical example is concentration of the chromatic number, appearing in Chapter 7 of Alon &amp; Spencer as well as many other places, such as Bukh's write-up.