How to Enable HTTPS / Secured Connection on Twitter
After Facebook and Google, now Twitter has upgraded security for its users. It now provides a setting to always use HTTPS. This allows for greater security and encrypted communication, particularly over unsecured Wifi connections.
To enable this setting, simply mark the checkbox Always use HTTPS in the Twitter account settings.
For sometime now, this secure feature has been available by visiting the address https://twitter.com. Now, this feature is available in the settings.
The mobile version of Twitter does not yet support this security setting. To use Twitter securely on the mobile browser, you have to visit https://mobile.twitter.com
Source : Official Twitter Blog