Is it normal to still be supported by your parents after graduating from college

Yes until you get on your feet. And it depends on what you majored in, how high a demand it is for what you are looking for and how flexible you are to take a not so ideal position.

Now granted this was a while ago - but I got a job about 2 or 3 months after graduating college. I lived with my parents until I felt able to stand on my own. I also worked for a temporary agency doing anything they would give me until I secured a permanent job so I would have some cash.

Your college must have a career placement office - have you reached out to them for some help? Have you been willing to take a basic job? Have you reached out to a recruiting agency? All thoughts that could help you secure a job.

I would think it depends on your major. I graduated in a different time when I had four job offers before I even graduated from college, because I majored in Accounting. This was in the early sixties. Our son graduated summa cum laude in Finance, and couldn't find a job. He earned a Masters in Communication from the University of Texas in Austin with honors, and now works for the university. We still give him some financial support.

Based on the economy today,it is normal. The #1 degree for finding a postgraduation job, is BS in Nursing from an excellent College/University.
If you start your nursing career at age 24 or so, you can expect to work for another 40 or 50 years ( depending on your health). Nurses (BSN) will always be in demand.

That's an excellent suggestion. I just looked up the starting salary for RNs in our area, and this is what I found.
Registered Nurse Salary in Silicon Valley, CA
In USD as of Jul 24, 2016
Neonatal Intensive Care Nurse in Silicon Valley, CA $91,000
Psychiatric Nurse in Silicon Valley, CA $81,000
Cardiovascular Nurse in Silicon Valley, CA $89,000
Labor Delivery Nurse in Silicon Valley, CA $91,000
Registered Nurse Salary in Silicon Valley, CA | Indeed.com