What if my daughter, Cathy is working part time? How do I file her income tax?

You don't include their income on your return. It would only go on their return if they are required to file one.  They don't have to file if they only have W2 income under 12,200 but can file to get back any withholding taken out in boxes 2 or 17.  But you don't get boxes 4 or 6 back.

To file a separate return for a dependent you need to set up a new account separate from the parents.  Online is only good for one return per account.  You can use the same email address for 5 accounts.  You can probably use the Free Edition or the Free File website

Https://ttlc.intuit.com/questions/1894512-how-do-i-start-another-return-in-turbotax-online 

Be sure on the dependent return you check the box that says you can be claimed on someone else’s return.  If your only income is W2 and under 12,200 you do not have to file a return except to get back any withholding taken out.

Filing requirements for a dependent

https://ttlc.intuit.com/community/children-dependents/help/do-i-need-to-file-my-own-taxes-if-i-m-a-dependent/00/26111

She would file her own tax return and indicate on the return that she can be claimed as a dependent.

Use the TurboTax Free File program if she is eligible - https://freefile.intuit.com/

If you are eligible you can use the TurboTax Free File Program which is free to file both a federal and state tax return. The Free File Program edition is a full featured personal tax program and on a separate website from the TurboTax online editions.

To qualify for free 2019 federal and state tax returns with the Free File Program, you'll need to meet at least one of these requirements:

• A maximum 2019 adjusted gross income (AGI) of $36,000; or
• You qualify for the Earned Income Tax Credit (EITC); or
• You served as active duty military (including Reservists and National Guard) with a maximum 2019 AGI of $69,000 (you'll also need a military-issued W-2).