Daughter started a new job in December 2023. Filled out a W4 (single and 0) however, her employer marked as exempt. Received penalty for estimated taxes. Is this right?

Yes, she has to pay the penalty. The employer made a mistake entering her W-4 in their payroll system, but that's between your daughter and the employer. She does have some responsibility to look at her pay stubs and make sure the amounts are correct, especially when starting a new job or when there has been a change.

The penalty doesn't have to do with last year's tax return or not paying estimates for this year.  The penalty is an "estimated" amount.    It's a penalty if you owe too much or for not paying in enough withholding during the year or not paying in evenly.  Even if you are getting a refund you can still owe a penalty.

The underpayment penalty amount Turbo Tax calculates on your 1040 or state is only an estimate.  Turbo Tax can not figure it accurately because they don't know exactly when the IRS or state will get your return and tax due payment.  It's very common and normal and expected for the IRS or state to bill you for more or send you a refund.