[Michael Butler]

Don't be too hasty in replacing your CPU, it's more of a GPU issue.
Try the same car/track combo offline with a full grid of AI, and start the race from the back of the pack. This should replicate the issue. Then turn down your graphics settings one by one starting with reflections, this is a major one. Anisotropic and anti aliasing are also big frame rate killers and shadows too.
What graphics card are you using?

This is exactly what I have been doing. I am now at a point with my settings that it doesn't happen with 24 cars but does happen with 25. 

Although it would still suggest I need to upgrade something.  I'm running a 1050ti

I realize you just bought that GPU...1050ti ( My son just bought the same) it's really considered an entry level gaming card hence the price. The GTX 980 would have been much faster but probably put out more heat......some modded tracks or cars will have a negative effect on slower components. I'm running the GTX 770 and i just recently starting experimenting why I'm dropping to 30 FPS at Road America while i normally run over 100FPS. Ive tried just about everything and i still get frame drops  and sudden 1 sec. freezes(just once a race)...I have the I7-4770 and it runs @ 90% playing AC. Don't give up on the 1050 ti. Kunos warns not to adjust GPU settings and let the application decide but i still get frame drops and mini freezes. Turning down graphics in the game is all you can do maybe. As long as your processor isn't pegged at 100% all the time, you should be ok.  Like i said before, some tracks or cars are better than others as far as FPS goes.