would a two stroke engine power a bike at 200cc
Yes, but that is huge if "bike" means bicycle. It would vibrate badly, produce terrible emissions from the burnt oil, and if talking about putting it on a bicycle it would (if geared normally to get the most out of the engine) go too fast to be safe on a bicycle frame, wheel and brake strength perspective.
In the US, many states also reclassify bicycles if the engine is over 49cc, meaning need for a motorcycle license, license plate, helmet, insurance, and other mandatory safety gear.
Stick with 49cc or less for a bicycle or a 4 stroke for a motorcycle.
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If your asking weather you could get a 2 stroke 200cc motorbike, yes you can, but they are VERY loud and annoying. if your asking if you can replace a 200cc engine with a 2 stroke, yes you can, anything is possible!