Water coming from the cistern is not capable of flushing the commode?
Flush toilets with the usual non-pressurized water tanks don't need significant water main pressure to function. The depth of the water in the tank is what provides the pressure. Low water main pressure will increase the time to fill the toilet tank, but once it's full the flush process will be the same.

In your case, it's likely there's something wrong with the toilet flushing mechanism, or the drain pipe.