Credit card limits are, for the most part, soft limits; sometimes, a credit card will allow you to charge a little over your limit. The large amount they allowed you to go over your limit is unusual, but not unheard of.
It is your responsibility to keep track of how much you charge on your credit card, not the bank. Just like with a checking account, you are supposed to keep track of everything you charge so that you always know how much you have spent and can pay it off.
Raising your limit will not help your problem; it would only make it worse. You have already charged more than you can afford, and they have already effectively raised your limit by $1200.
I realize that this situation is tough, but fortunately, you have a learning opportunity here. I recommend you resolve to stop using the credit card in this way, and work toward paying off your debt to zero. At that point, treat your credit card as if it was a debit card, and only charge what you already have in the bank to pay off right away. (Or, just use a debit card and get rid of the credit card.) Learning to do this now will save you lots of money in interest.
If you don't learn to do this, you will find yourself in even more debt in the future, and it will be even harder to dig yourself out.
If you need some more help on getting out of debt and learning to budget your money, I recommend the book The Total Money Makeover by Dave Ramsey.