Do Fighter Jets Experience Turbulence?

In general - at least compared to most non-G.A. civil aircraft - military fighters have quite a low, not high, wing-loading. Low wing-loading aircraft ride relatively worse in turbulence, so turbulence would typically have a greater effect on military jets. The F-104 and F-105 are rare exceptions.

Do private jets feel more turbulence?

Private jets are less likely to encounter turbulence than commercial aircraft, as they have a quicker climb and usually cruise above turbulence. But when private jets do experience turbulence, their smaller size means that it may feel more violent to passengers than it would in a larger commercial jet.

Can fighter jets fly in bad weather?

Yes, just occasionally the weather is too bad for any aircraft to fly safely. Modern weather forecasts can usually predict this, and this means that aircraft stay on the ground when this is likely to occur.