Kristina,
Family Search has not been giving me any problems during the Connect-a-thon. They say they support the current and the last versions of Safari only. So make sure you are using one of those.
Another suggestion which may or may not help but it is worth a try (if you have not attempted it yet). In your browser settings, try clearing your browser history of everything.
WARNING - If you clear everything in your history, you will need to re-enter any saved passwords and signin information saved. You can choose not to clear everything and select the categories you want cleared. However, one of those items saved might be causing this issue.
Shut down your browser completely. Reopen and try accessing Family Search.
If you are using a Pop-Up Blocker, this can interfere with the functionality of Family Search as well. You would need to allow pop-ups from Family Search.
If this does not work, I hope that someone with Safari on iMac will help you soon!