Bicycle:
Road tyres are thin to reduce rolling resistance so the rider can go faster for longer. Traction is not a big problem with about 1bhp peak to handle.
Off-road: Tyres are bigger to absorb bumps and to spread the load of the rider and bike so they don’t sink into the soft stuff.
Motorcycle:
Road tyres are sized according to the power and weight of the bike. A 125 which weighs 100kg and has 15bhp can happily apply power through a turn without losing grip. A 1000cc with 200kg and 180bhp needs a lot more tyre area to do the same. A large bike could corner on the smaller tyres but applying the power becomes problematic.
Off-road. Tyres need to be large enough to spread the weight and small enough to dig into the ground and to steer. It’s a trade off between digging holes and going nowhere or acting like a sled, spinning and going nowhere. Traction is always an issue but it’s not one that a larger tyre can solve.
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