Enter the $5,000 in box 10 on your W-2 (I assume that it is already there) for your 2019 tax return..
Then, when you are asked in the Dependent and Child Care credit section how much you actually spent in 2019, enter the $3,345.
When you enter your tax year 2020 return (i.e., next year), you will be asked if you made any payments from the 2019 FSA funds carried over to 2020, i.e., payments permitted by the "grace period" (where you can carry-over FSA dollars for the first 2 1/2 months in the next year). You will enter the $1,655 (or whatever you actually spend of carryover).
TurboTax will complete the form 2441 accordingly.
As for your care provider, nearly all individual taxpayers operate on a cash basis, that is, you report income and expenses in the year that you receive the income and make the expenditures. Thus, your care provider will report the $3,345 that you actually paid in 2019, and will report the other $1,655 (or whatever) as part of her/his income in 2020.
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