No. The 20% that was withheld at the time of the distribution is federal income tax withholding. It is not a penalty — it is a prepayment toward any income taxes you owe for the year. That amount is included in the total taxes paid on your return and is applied against your overall tax liability.
The additional 10% charge is an early withdrawal penalty, which applies because the funds were withdrawn before age 59½ and no IRS exception applies. This penalty is separate from regular income tax and is calculated on the taxable portion of the distribution.
Refer to the TurboTax articles: An Early Withdrawal From Your 401(k): Understanding the Consequences and Video: A Guide to 401k and IRA Early Withdrawal for more information.
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