Did the employer withhold tax from your pay? If not, then you were an independent contractor, or at least they were treating you as an independent contractor, and that's exactly how you should file. The company that paid you as an independent contractor was not your employer. They were your customer, and you were in business for yourself.
Did they give you a Form 1099-NEC? To file as an independent contractor without a 1099-NEC you will have to upgrade to TurboTax Premium Online, or use any edition of the CD/Download TurboTax software.
If they did withhold tax, they should have given you a W-2.