Goto the URL opera://settings/content/all, type x.com in the search field and click "Delete displayed data". Then, repeat for twitter.com for good measure.
Then, goto the URL opera://settings/clearBrowserData and clear "cached images and files" for all time.
Then, open up a new tab, type https://x.com/ in the address field and press enter to see if it loads and then log in. Update the URL for any X/Twitter bookmarks and speed dials you have that use twitter.com too for good measure.
If none of those the trick, it could be one of your extensions at the URL opera://extensions interfering.. Since things work fine in an incognito windows, it'd probably be one of your extensions that isn't allowed to run in incognito mode.
You can also test with Opera's adblocking and tracking protection off at the URL opera://settings/privacyProtection if needed.