Important: the answer below assumes that you and your GF are both the biological parents of the child and the child lives with you more than half the year. If the child is only the biological child of one of you, only that parent can claim the child as a dependent on their tax return.
My guess is that your girlfriend should claim the child, but only you can test the different possibilities and know for sure. If using turbotax online, you will need separate online accounts for each taxpayer, and you will pay separate fees. Your girlfriend can use the Tax Freedom version, if the regular "free to start" version tries to charge a fee for certain forms. If you use the desktop version installed on your own computer, you can make as many test returns as you want, they are stored as different documents.
Your girlfriend can claim the childcare expenses as well, unless you used a workplace FSA to pay them. (If you did use an FSA, then by not claiming the child as a dependent yourself, you will increase your own taxable income, which is why only you can really test which scenario is best in your case.) She can say the expenses were a gift from you to her and then she paid them. (This should survive an audit, if it comes to that, although it would be better if the payments came from a checking account in her name or a joint account.)
Whatever you do, be sure that the person who does not claim the child as a dependent does not list the child at all on their return, and the person who does claim the child answers "No" to the question about an agreement to share custody. This question only applies to parents who are divorced or separated and living apart. And make sure that if you make test returns, you triple check to make sure the returns you actually file only contain the correct and final information you want to include.
