Thank you for your question.
Unfortunately with regard to federal income taxes, those are your responsibility whether the employer took them out or not.
The IRS position is that the tax payer needs to be checking pay stubs to make sure the employer is withholding taxes.
The reason is, that the individual tax payer has a personal legal requirement to pay estimated taxes to the IRS quarterly. But since most people would not do this voluntary, the IRS and the congress passed a law creating an agency relationship between employers and the IRS (and a few others such as real estate closing agents, etc), to make them withhold taxes from employee paychecks and pay them quarterly to the IRS.
As individuals we meet our quarterly requirement by submitting the W-4 to the employer. Even in situations where the employee does not submit a W-4, the employer is then required to withhold at the default rate of single and zero.
IN your case the employer did not fulfill his or her agent responsibility by withholding properly. He is subject to a fine, but really, a small fine it is. So you can report him to the IRS for not withholding, and based on that try to get the IRS to abate the interest and penalty for non-payment of those first two quarterly estimated tax payments.
If not, then you could go to small claims court to sue the employer for reimbursement of the penalties and interest that you will likely be assessed for non-payment of those first two quarters.
You may not even have to pay interest or penalties if the tax you actually did have withheld and paid is close to your actual tax liability for the previous year.