The temporary increase was for plan years starting in 2021. For most people, that would be January 1, but for your employer, that is July 1, 2021. So the increased limit applies to your entire plan year July 1, 2021 through June 30, 2022. You have nothing to worry about. (When your new plan starts July 1, 2022, you will revert to the original $5000 limit.)
As to whether Turbotax will handle this, it should, but I haven't tested it. Post back if you think it is not being handled correctly. You should be asked for the dates of the plan year if it is not Jan 1-Dec 31.
[Edited]
Your maximum election to an FSA beginning July 1, 2022 will be $5000. The IRS notice that expanded the eligible contribution specified that it covered any plan year starting in 2021. Most plans start Jan 1, but if your plan started July 1, 2021, then you can elect $10,500 for your 21-22 plan year, and you will go back to the $5000 maximum election for the 2022-23 plan year.
However, being able to elect a certain contribution is disconnected from being able to actually exclude the benefit from income tax, because the exclusion on your tax return is $10,500 for the 2021 calendar year, and reverts back to $5000 for the 2022 calendar year. This can create a situation where you can choose a benefit that is larger than you can actually claim on your tax return. If this happens, the excess is added back to your taxable income, but there is no additional penalty. The formula and situations are complex and involve carryovers and grace periods. If you have been using your benefit all along, you are probably in the situation of example 3 from IRS notice 2021-26 (see below). You would only be able to exclude $5000 on your 2022 tax return, even though you might actually have $5000 of 21-22 benefit left over plus $2500 of 22-23 benefit.
So the answer to "what can I elect for the plan starting July 1, 2022" is $5000. The answer to "will it all be tax-free on my 2022 tax return?" is probably not, but it's complicated.
