It is critical that both the bowl AND the tank get properly filled during a flush/refill cycle. If the bowl starts out less than full, it will not flush well. Flush the toilet, note where the water level is in both the tank and bowl after it's refilled. THen, take your bucket and slowly pour water into the bowl. If you can get the water higher in the bowl (it will take a few moments after pouring to stabilize), then the bowl is not being filled, and the flush can be poor. There are some aftermarket filler valves that are adjustable to get the balance right...you may need one of those.
THen, if your toilet has a siphon jet (a hole down near the bottom of the bowl), this may be clogged. If it has one, this creates a jet of water to start the water leaving the bowl. If that is plugged up, it won't flush well. On some toilets, they used a float on the flapper valve...plumbers have reported finding several of those clogged inside the internal passageways. If your toilet design uses one of those, and you seem to have lost one, that may be your problem.