Check to make sure you can take a bicycle down the Turkey Path legally. You may be able to, but some trails are marked no bicycles.
If allowed, it’s still not a great plan. The trail is steep, narrow, and full of switchbacks. Unless you’re a very accomplished mountain biker, you’re gonna be walking that bike down, and probably wishing you just biked the rail trail in from Blackwell or Darling Run. If you have a good 4WD, and I mean a good one…Legit 4WD or a Subaru, Tiadaghton becomes a viable option, and even though I swore I’d never go down that road again (I have a 4Runner and a Tacoma) I’d do that before trying to take a bike down the Turkey Path.
Look up some hiking trip reports on the Turkey Path. That should paint the picture for you.