Normally I use Chrome on my Mac. I just tried on my Safari on Mac OS, no problem at all. The new window opened with the Twitter link right away. It is strange you are having these issues.
I suggest trying it on your smartphone. I just tried on my iPhone on Safari, it worked fine for me.
Tried doing the twitter referral with Firefox on my iMac. Worked. Applied for a card for my business today online. Got the pending screen. Will call on Tuesday 9/5 to try to speed up the process. I have some big bills coming due soon that I'd like to use this new card to pay with and get the $5k spend on in the first billing month.
Update: I called the automated line 9/5 and was approved. I called a human to confirm it. the card is on it's way. This was the first time I applied through a referral link so I'll be interested to see if the referring party (me) gets the bonus 20K points.