If you wait long enough, Telstra will accidentally delete these for you along with everything else.
Not aware of the Webmail software Telstra use, but I wouldn't be surprised if all your e-mail, including your sent e-mail, sits on their servers. You can prove this by opening up your e-mail account on another PC and checking for the latest activity.
I doubt if sent e-mail slows down your computer, user mail boxes of type "maildir" are now common and are structured in such a way that they are very fast and efficient, however I wouldn't discount Telstra living in the past and using the old "mbox" mail format on their servers where everything lives in the one file - that's so last century. If that's the case then everything will be slow if the mbox file is big.
As for deleting more than 50 messages at a time, unless Telstra have provided an API to your mail account and you know how to interact with it then you are at the mercy of the features that their UI provides. Perhaps there's an "archive and delete" option that could help?
Sorry I can't be of more help, other than to say I know enough about Telstra to steer well clear from them.