Strange Link Problem
The cause of this issue with a 90% likelyhood are your craft settings

Before you try any of the below steps, try clearing your cache first!

Sometimes some leftover cache is all that needs to be removed to fix an issue...
rm -rf runtime/* or ./craft clear-caches/all

3 Settings which you may have set wrongly:

All of the below mentioned settings should basically be pointing towards my_current_website.com, a reference to that or something more relative like a simple /.

There are 3 Places where you should check if you have my_live_website.com set as your current host/site.


.env's SITE_URL, BASE_URL or anything else that has  SITE or URL in it's name
config/general.php's siteUrl Value(s)
Your Craft Control Panel Settings, my_current_website.com/admin/settings/sites (URL may differ based on config/general.php's cpTrigger value)