If I claim my babysitter expenses, will TurboTax 1099 her and what is the threshold amount?

No Turbo Tax won't send her a 1099.  But the IRS will check if she reports all her income.  You don't have to give her a 1099.  You are not a business.  She has to report her income from her own records.  

No TurboTax will not "1099" your babysitter for you. 

If you are using the childcare credit you have to enter your babysitter's SSN or Tax ID.  It is then the babysitter's responsibility to report their income on a tax return.  The IRS matches up the amount you said you paid with the amount on the sitter's tax return.

Did the sitter work in your home or the sitter's home?  If it was in YOUR home and you paid more than $2400 in 2022 you were supposed to treat the sitter as a household employee and issue a W-2.   If the sitter worked in their own home or had less than $2400 for taking care of your child they have self-employment income to report.