Do you have good "before and after" examples of websites that have undergone user experience optimisation?
What were the key differences (include stats if possible) as a result of it?
I have never worked on this website personally, but Amazon is a great example. If you use an online tool called The Way Back Machine (its like a visual internet archive) you can see what their website looked like when they first launched. It was dreadful (as most websites were in the early to mid 90s).
Once Amazon started selling more than books, their site quickly got a lot more "noisy" both in the amount of navigation and products to filter through.
After several iterations their design team created an optimal checkout paradigm (the 1-click checkout for signed in users). They went so far as to file trademarks and patents around the process. By lowering the barrier to checkout for frequent users they made impulse buying much easier (for better or worse on our bank accounts).
There was a second combination that become the knockout punch for online buying. It was the algorithm they created to display products other people bought (or ones you previously looked at) next to what you were about to add to your cart. This can get a bit creepy at times, but it lead to much higher conversion rates. I don't have specific stats on it, but this has become a pattern on other ecommerce websites as well.
As a bonus, in the Search world, I think that is why Google as a late arrival to the Search game compared to (AOL, Excite, Lycos, Yahoo, etc.) was able to disrupt them. All the other search engines at the time had very busy page layouts with much too much taking the focus off of search. Google came on the scene and only put their name and the search field. This seemed contrary to all rivals but it worked brilliantly.
This is because sometimes being a service that does 1 thing extremely well, is much more beneficial than trying to be all things to all groups of people.