I doubt it. Tor Browser has been design exactly to stop such tracking attempts. It would appear to me that you want to use a regular browser with Tor (Tor the proxy not Tor Browser.)
To make a regular browser use Tor, you'll have install Tor and configure a browser to use it as a proxy.
A Word of Warning
Based on your question, I assume that anonymity isn't the reason you want to use Tor. You should be aware that in this setup does not provide any anonymity, even your real IP might me leaked. Regular browsers are simply not design to guard against this.
1. Install Tor
on Linux: Just install tor via package manager. If you want a more up-to-date Tor version take a look at the expert guides.
on Windows: Duckduckgoing for it indicate that you need the expert bundle.
on Mac: The wiki appears to have some instructions.
2. Configure Firefox
Open the preferences then go to Advanced → Network → Settings and change your proxy settings like this:
Should also work with other browsers, you just need to find the proxy settings.