Your employer does not decide or determine whether or not you are exempt from paying taxes. Since you are not exempt from income taxes, you should give your employer an updated W-4 that reflects this information. Your employer should be withholding income taxes and those taxes should be reported in Box 2 of your W-2. If your employer has not been withholding and you earned over 12K and do not have a qualifying dependent, you will probably owe income taxes.
You can prepare a new form in TurboTax to give to your employer by typing in W-4 in the search box and using the 'Jump to' link. Or you can request a blank form from your employer and update it there.