Bottle-necking can and does always occur on either the CPU or GPU in any configuration depending on the game conditions and visual settings (unless throttle is induced, like Vsync, creating a forced intentional bottleneck). There's nothing particularly wrong with this in and of itself.
What resolution and detail and post processing settings are you using and what sort of FPS are you getting? If the FPS is fine but the GPU is not being fully utilized, the answer is to simply turn UP the visual quality until the GPU is saturated with work.
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The 95W AMD OEM HSF is garbage. If you're using that you should swap it for something with heat-pipes. Doesn't have to be big or fancy. Even a $20 unit with 3 heat pipes and a 90mm fan is an enormous upgrade over the stock block of aluminium that AMD ships these with. The CNPS5X or A11 or AR02 or TX3 or Vortex Plus, all<$30 and all will keep an FX-6300 well below thermal margins at stock clocks (in fact, they will even support a bit of overclocking in most cases).
What motherboard is this on? Provided you have something of decent quality it might be worth doing some conservative performance tuning. IMO the FX-6300 really hit pay-dirt when overclocked to ~4.4ghz+ (the practical limits tend to be in the 4.2-4.8ghz range depending on available cooling, VRMs, and specific CPU voltage/leakage characteristics).