Is there a place on Schedule C expenses to enter self-employed state tax payments made in 2018 on the C form or elsewhere?

Any quarterly state tax payments would get entered in the estimated payments section of TurboTax.  See the steps below to be brought to this section of TurboTax.  Be sure to enter them as STATE estimated tax payments.
Where do I enter my estimated tax payments?

 

Thank you, Julie. What's confusing to me is that there is the possibility of entering the ES payments paid in 2020 in the schedule C section for business deductions -- OR on the personal tab page under deductions. ???

 

You will post the estimated payments to the personal section of your return. 

 

Your Schedule C business flows directly into your personal tax return.  So that is where the payments should be posted.

 

The Estimated tax numbers are posted to:

1. Deductions & Credits
2. Estimates and Other Taxes Paid
3. Estimated Tax Payments
4. State estimated taxes for 2020 (Form 1040ES)

 

So, does that mean I post these payments in both sections? Sorry, still not clear.

No, you will only enter them as indicated by JohnB5677 above.

 

You will not enter them anywhere on your Schedule C.