Received signing bonus in 2018, returned it in 2019. How to handle for 2019 taxes?

If you can't get the company to "correct" your W-2 for 2018, your only option is a deduction under IRC §1341 based on the “right of claim” doctrine. This rule allows taxpayers to deduct an amount that was previously included in income as long as it’s more than $3,000 and shouldn’t have been included in the income calculation because the taxpayer actually didn’t have the right to it.

This method allows taxpayers to reduce their tax liability in the year when the bonus was repaid by a credit in the amount of the tax difference. It’s calculated as the larger of:

• The tax paid in the bonus year with the bonus payment minus the tax paid in the bonus year without the bonus, or
• The tax paid in the current year without a de­duction for the repayment minus the tax paid in the current year with a deduction for the repayment.

See this answer from Critter to a question with a similar issue:

Repayment over $3,000.    If the amount you repaid was more than $3,000, you can recover the tax you originally paid using one of two methods. You can do it as a deduction on schedule A or a tax credit on line 73 of the 1040 form if when you included the income in a previous year, it appeared that you had an unrestricted right to it.

Figure your tax under both methods and compare the results. Use the method (deduction or credit) that results in less tax.

Method 1.    Claiming a deduction for the repaid amount. You deduct it as a miscellaneous itemized deduction on Schedule A  line 28.

This is the flow in TurboTax: 
Federal taxes > Deductions & credits > I'll choose what I work on > Other deductions and credits > Other deductible expense>Go thru to third screen> answer YES to "Did you have any other deductions that are not subject to the 2% limitation?"
Enter the amount you repaid under, "Claim of Right Repayment (Only if over $3,000).

Method 2.    Figure your tax claiming a credit for the repaid amount. Follow these steps.

1.      Figure your tax without deducting the repaid amount.

2.      Refigure your tax from the earlier year without including the income that you had to repaid.

3.      Subtract the tax calculated  in (2) from the tax shown on your return for the earlier year. This is the credit.

This is the flow in TurboTax (This can only be done in the Forms mode of the Desktop version of TurboTax)

Go to FORMS (top right of main screen) > Form 1040 > Scroll below line 72 to "Other Payments and Credits Smart worksheet"
Please enter the credit o line D. 
This credit should show up on Form 1040 line 73