Your description is a little hard to follow because your terminology is not very precise.. Some suggestions:
"turning over" means the starter is spinning the engine over in an effort to start.
"turn on" means flip an electrical switch to the on position.
"start" means the engine begins to run.
You should know that the ignition system on these bikes is entirely separate from the 12 volt battery and electrical system. it is powered from its own power coil located in the alternator stator. Therefore the condition of the battery will only affect the starter operation but not ignition. If you can get it running, the ignition will continue to operate even with a dead battery.
The alternator needs to be spinning pretty fast and for a long time to recharge a weakened battery, so if it is hard to start, and you are making many short trips, the battery may run down completely without ever recharging. You should get a battery tender to recharge your battery and keep it topped up when parked.
From your description it is difficult to say where the issue may lie. The fact that you say it ran perfect with a new battery and then began running poorly suggests the issue might be in the ignition module, although these can be expensive and hard to find and I would want to be sure it wasn't something else before changing this.
2004 Rebel 250, 2003 BMW K1200GT (roadburner), 2004 BMW R1200GS(all purpose),
1973 Norton Interstate (in a box in the basement)
1968 Triumph Bonneville