The rules for deducting car expenses are in chapter 4 of publication 463. Pay attention to the difference between the "Actual expense" method and the standard mileage method, especially regarding the records and documentation required.
https://www.irs.gov/forms-pubs/about-publication-463
Then, if you are a W-2 employee, your costs are not deductible on your federal return. You can list them on in Turbotax and they may carry over to a state deduction, depending on your state and your other tax facts. If you are self-employed, you file a schedule C to report your business--all your gross income, and all your expenses, and you pay income tax and self-employment tax on the net profit. Vehicle expenses are listed on schedule C along with your other expenses.
If you are a delivery driver and a W-2 employee, your company can (but is not required) to reimburse your expenses tax-free, but if not reimbursed, you can't deduct them. However, if the extra expenses drop your net income below minimum wage, you can complain to your state's employment regulator. I am aware of a business in New York that was required to pay several years back wages to their delivery drivers because they required drivers to use their own cars and did not reimburse them, and some of the drivers made less than minimum wage after taking car expenses into account.
