That is your problem, wrong battery.
Bike Batteries are sold with 2 important measurements:
CCA (Cold Cranking Amps) and AH (ampere Hours)
The definition of the CCA of a motorcycle battery is the amount of current a given battery can deliver for 30 seconds at zero degrees (Fahrenheit) without dropping below a specified cut off voltage (manufacturer-specific, but usually 10.5 volts). The value of CCA will change with battery temperature, CCA increasing with higher temperatures. Therefore a 180 CCA will allow 180 amps flow for 30 seconds at zero (Fahrenheit) before reaching the cut off voltage
The definition of AH is the amount of charge in a battery that will allow one ampere of current to flow for one hour. Therefore a 10 AH battery will flow 1 amp for 10 hours before going completely flat. On the 2002/2003 Fazer 600s that would be the same as letting the headlights on high beams on for one hour.
The battery you have in your bike is a 9AH model with only 85A CCA and is normally used in small capacity bikes or kick start only larger old bikes.
The OEM battery for the FZS600 is a GS GT12B−4, which is 10 ampere/hour,180A CCA.
My advice is dump the battery and get the correct size one.