Usually, a timed out session has to do with inactivity on the website, not on the computer. It's a safety feature, so if you walk away from your computer for a while, somebody can't just come in and take over your e-mail account.
Just be sure to be active within your e-mail website every few minutes to avoid being timed out. Alternatively, you can be sure that you're active on the e-mail website before you start composing an e-mail.