Q: I lost my iphone. How can i delete my text messages?

You can erase your device threw icloud's findmyiphone.

iCloud: Erase your device

If you have iCloud configured, to delete use

https://www.icloud.com

Read here: iCloud: Erase your device

You can't erase just text messages remotely but you can erase the entire device if it is still online. See below from If your iPhone, iPad, or iPod touch is lost or stolen - Apple Support

Erase your device. To prevent anyone else from accessing the data on your missing device, you can erase it remotely. When you erase your device, all of your information (including credit or debit cards for Apple Pay) will be deleted from the device, and you won't be able to find it using Find My iPhone. After you erase a device, you can't track it. If you remove the device from your account after you erase it, Activation Lock will be turned off. This allows another person to activate and use your device.