https://chrome.google.com/webstore/search/click%26clean
I use this Chrome extension. There are a few settings to go through to get it to delete data on exit. One small downside, you have to close the browser window (red x in top left) for it to work before exiting the browser (cmd Q). Better than forever waiting for a native feature.
https://addons.opera.com/en/extensions/details/download-chrome-extension-9/?display=en
You also need this Opera extension in order to use Chrome extensions, if you weren't aware.