The form says detailed info. must be submitted with my State return when claiming a medical expense for work, but TurboTax isn't giving me any opportunity to add it.

A medical expense for work is considered an Employee Business Expense. Employee business expenses have been suspended on Form 1040 Schedule A Miscellaneous Itemized Deductions Subject to the 2% Floor starting in 2018. 

Some states do allow Employee Business Expenses on their state return. If your state allows this expense, you would enter the information on the Federal return. It will carry to the state return.

To enter the expense on Form 2106 on the federal return:

• Locate the Search Magnifying Glass in the upper right hand corner of the screen.
• Enter Form 2106 in the search area and hit Enter.
• Select Jump to form 2106.
• Enter the information on the employee business expense information on the screens.