I made a Survey in Survey 123 Connect and I was wondering if it is possible to automatically save the form and open a new empty form after clicking/pressing on the check mark in the right bottom of the screen.
I want to use the survey for counting birds so I have to submit the data fast.
No, this is not currently possible. If you want to see this as a new feature or enhancement in the future, I suggest you create a new idea on the ArcGIS Survey123 Ideas website.
Alternatively, have you considered using a repeat (related table) in your survey. The parent layer could contain just your date/day, location and user information. And then the related table could include information about the birds sighted, and you add a new record for each bird seen. This would allow you to add new records quickly in the same survey with details without needing to create an entire new survey.