You will enter state estimated payments on the federal return.  For each payment made, you will indicate the date and the state that you paid it to.   Estimated payments will then flow to the appropriate lines on each of the state returns.  

State income tax returns can vary considerably from one state to another. TurboTax will put your estimated payments on the correct lines for each state that you file.  The estimated taxes will be calculated when determining the amount of refund or the amount to be paid. 

Individuals that make multi state estimated payments are relatively rare.  Most people working in multiple states are W2 employees and their state taxes are withheld from their pay and recorded on their W2 form.

TT does not give you the option of  "multiple states" in the estimated taxes section.  If you leave the state blank, TT will apply your entry to your home state.

The "multiple states" option usually comes up in the dividends and interest section (box 11 of form 1099-DIV or box 8 of form 1099-INT).  Since only your home state taxes federally tax free interest, entering multiple states is OK.  Otherwise,  you going to have to ask your broker/mutual fund co. for a breakdown (usually available at their web site with some search effort). 

Did you make estimated payments OR payments with extensions?  They are different.

If you made multiple state extension payments, you will need to enter the total amount of state extension payments you've made. The program will enter this total on your federal return but you will need to go to each state return and enter the extension payment in the state interview for it to be accurately reported on yout state return.

For Online versions:

Although tedious and buried in TurboTax online, you can enter in payments made with extensions for multiple states.  You have to go into the interview for each state and get to the section regarding filing an extension.  Once there, you will need to say you are going to file an extension and then it will eventually trigger a screen asking for the payment amount you will make with the extension.  Once the amount is entered it will populate on the corresponding form and you should see the TurboTax refund meter adjust accordingly.