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The answer to width is rolling resistance.
Tires stick to road which is great for accelerating and breaking, but it makes maintaining your top speed harder. This doesn't matter if you have a big engine, but it makes a difference if you're cycling.
For a road bike, you want to minimize the contact area of the tire with the road so you have very narrow tires and inflated a lot so that they don't deform much under your weight.
The type and width of tires changes depending on what you're riding on. For off road you have wide knobbly tires that will catch in mud and push you forward. For riding on a beach you have very wide smooth tires for traveling over sand.
The diameter is about stability. If you have small wheels turning fast it effectively lowers your center of gravity due to gyroscopic effects (this is a massive over simplification of the physics) big wheels turning more slowly results in a higher center of gravity, which is more stable.
Cars have four wheels and don't need to care about this.
It's slightly different:
As the top speed of a road bike makes air resistance (drag) an important factor, road bikes use narrow tyres, resulting in a smaller silhouette area than a wide tyre (on a rim of the same diametre) would have.
As the rolling resistance increases with the length of the contact area, with the same internal pressure (inflation), i.e. same area of contact, narrow tyres have a higher rolling resistance than wide tyres. Thus, to (over-)compensate and decrease the length of the area of contact the internal pressure of road bike tyres is much larger than of normal, wider tyres.
As a result, narrow tyres of road bikes have smaller drag and due to over-compensation by inflation an even lower rolling resistance than standard bike tyres.
Edit: This over-compensation is possible for road bikes, as hard surfaces (asphalt, concrete, pavement) allow a high surface pressure.
For tyres of dirt bikes not sinking deeply into soft ground (gravel, soil), they need to have a low ground pressure, i.e. a large area of contact and low inflation and thus, cannot be narrow, but have to be wide.
you sure about it chief? bigger wheels also have larger rotational inertia
Yeah, try riding a folding bike with small wheels, you wobble a lot more.
It's not about the magnitude of the rotational inertia (which is roughly the same or maybe a bit higher on larger wheels that are probably heavier), it's about the center of the gyroscopic effect. It's at a greater height if you have big wheels, and closer to the ground if you have small ones.
A low center of gravity is more stable than a high one.
Gyroscopic effects do exist on bikes but they are not the main source of stability - it's the fork geometry, which tends to straighten the handlebar
The rotational moment of inertia increases a lot due to the larger diameter, J ~ m r^2. Even if the masses were the same, the relatively heavy tyre and the rim are further away from the wheel hub.
Edit: If we neglect the wheel hub and the spokes, the mass of the tyre and rim scales with the radius, m ~ r, and thus J ~ r^3 .