You said "mailed my W-2", but hopefully, you meant you mailed the tax return and the W-2.
If you filed by mail, did you print, sign, date, and physically mail it? TurboTax does not mail it for you. Hopefully, you mailed the Federal return to the IRS, and that you mailed any state return (if applicable) to your state tax agency. They have to be mailed separately to each place.
Normally when mailing things to the IRS or state, it's a good idea to use a method that has tracking such as certified return receipt, priority mail, etc.
If you didn't use tracking, then you won't know if/when the IRS gets it for some time.
After 4 weeks from when you mailed it, if you are due a Federal refund, you can use this tool at the IRS website to check on the refund:
Enter your E-mail address. We'll send you an e-mail with instructions to reset your password.