(Gas) Motorbikes are indeed illegal in Suzhou. They aren't sold here anymore, and the city doesn't issue licenses for motorbikes. This is apparently happening in many places in China.
The rationale is that gas bikes are too powerful and fast, and that people who drive them will go at speeds that make any collision or accident truly dangerous. (The late, lamented Rocket, my now-stolen e-bike, fell into the same category...)
I've kinda wondered about the angle of motorcycles making people too mobile in traffic and narrow spaces, and capable of too much range, and too hard for the police to catch...but that's just me.
You CAN buy used
motorbikes legally in Suzhou (for now), and drive them until their current tags expire. When they expire, the city won't renew them. The folks you see driving gas bikes- and doing so legally- did it this way. But their days are numbered.
A lot of folks are also just going to Shanghai or elsewhere and buying motorbikes, and driving them around flagrantly illegally. The expats rely on the fact that foreigners rarely get hassled for riding gas bikes (scooters, anyway...), although this is proving to be increasingly less true as local anuses (ani?
) reach their full Olympic clenched pucker. The homeys rely on local connections and/or knowing well where all the anti-motorbike traffic checkpoints are.
I dunno 'bouts no quad bikes (wouldn't that be a go-kart or a golf cart or some such?) but I DO know the local spud who drives the sidecar motorcycle (he's a member here, and a close personal friend), and so do the cops. That contraption is indeed way too cool for the likes of him, but it's howlingly
illegal here, and some day The Boys In Black are gonna chase him down like the mad and mangy dog (half Irish Setter, half French Poodle) he so truly is. They've come close several times now. And unlike a gas scooter, the thing is extremely conspicuous. Especially with me, wearing my pith helmet, in the sidecar.
Anyway, you'll notice you don't see the big wuss driving the thing around very often anymore.
I'd love to have a gas scooter in Suzhou. All the great things about a good e-bike, without the limited battery range...
But legally it's not really very doable in Suzhou...and life here is nuts enough already.