How can I use the desktop version of Google translate on my iPhone?
I think what you want is in the Google Translate ios app. At least the phrasebook.

Here's Google's documentation that talks about how to use the phrasebook (save translations) on different platforms.


  Save translations - IOS
  
  Add a word or phrase
  
  
  On your iPhone or iPad, open the Translate app.
  Translate a word or phrase.
  Next to the translation, tap Star translation.
  
  
  See &amp; delete translations
  
  
  On your iPhone or iPad, open the Translate app.
  At the bottom, tap Saved.
  To remove a word or phrase, tap Star translation.
  
  
  
    Note: Your app phrasebook saves on the app only. You won't see it on other devices or in your mobile browser.