Short answer, yes, because taxes are your responsibility. You either filled out the W-4 wrong, or it was your responsibility to notice that the withholding did not match what you requested and correct it with your employer. You might be able to complain to a state employment regulator, and there might be a state law that would penalize the employer for not following your instructions, but there is no recourse with the IRS to argue your employer made a mistake.
As noted, if you make an estimated payment by January 15, or if you file your final tax return and pay in full by January 31, that may mitigate your penalties. Getting caught up as soon as you can will reduce the interest portion of the late fee even if the IRS still assesses a penalty portion.
Also, I would not allow Turbotax to calculate a penalty. My suggestion is to simply file and pay your taxes without including a penalty. Wait and see if the IRS bills you. You can then ask for an abatement (waiver) of the penalty for cause, or if this is your first time. It is easier to ask for an abatement when they bill you, than paying a penalty with your taxes and then asking for an abatement later.
