TurboTax, give me 4 bills to pay my Income Tax Return, now the Illinois State asking me to pay the complete value, with penalty, without deducting the first payment

[Critter]

No ... read those forms again ... those are NOT payments for the 2018 balance due they are ESTIMATED payments for the 2019 tax year.   

[VolvoGirl]

Oh no.  Those were not for your current tax due.  Those were the 4 quarterly Estimated payments for NEXT year.  They are optional to pay.  They are so you don't owe so much again next year.  See, do they say 2019 on them?  Don't forget to enter them into next year's 2019 return to get credit for them.

Sorry you still owe the full 2018 tax due.

[Critter]

Check the printout or PDF of your return; look for the state cover sheet with the Turbotax logo.  If you owed tax, it will show the payment information and how/when you decided to pay.  Read the state payment instructions carefully since most states cannot be paid from within the TT system and requires additional steps.
https://ttlc.intuit.com/questions/1900884-what-are-my-options-for-getting-my-state-refund-or-for-pay...

Sadly the first payment you made using the estimated form was applied to 2019 .... you may get them to release it for the 2018 tax year ... call the number on the notice you got an see if it can be done.