Easiest - relay them
- if the texts are not confidential, text them to a friend that does have email on their phone and ask them to email them to you 
A bit harder ...
- get access to your email on your phone. Email on your desktop communicates with an email server (gmail, hotmail, yahoo, whatever). You can configure the mail app on your iPhone to talk with the same email service so you can also send (and read) email on your phone. The Mail.app is standard in iOS, you don't have to download it from the store!
Also a bit harder
- you didn't say what type of desktop system you have, but if it's a Macintosh, you can set it up to also receive (and send) text messages that are shared with the same texts that are on your phone.
Lots of other options
- You can install Signal (or Telegram or Whatsapp) on both your phone and desktop and copy/paste the texts to a message to yourself
- Lots of applications have shared files between iPhone/desktops
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