A TurboTax employee on the phone had me download Cobrowse so he could see my screen, and now I'm a little weirded out. Is this legit?

Yes, it is completely legit.  It is a tool so the person on the phone can see your screen to more efficiently help solve your problem. When you were done with the session you can log out of it, although the phone agent probably already did.  

I understand what the tools is, but the last time I called with a question, I shared screens via a browser-based application. Having to download and install something felt like a leap of faith there. Thanks!

Depending on the problem you were having, sometimes the co-browse is necessary because of the limitation of ScreenShare.