How to increase output power of this simple CD4047 based inverter
In all probability you have insufficient voltage on the output to run your CFLs.

You are driving the 12-0-12 transformer with a 12 V square wave. That is giving you approximately a 228 V square wave on the output side. 
The domestic 230 VAC used to power a load is about 322 volts peak and since almost universally there is a bridge rectifier in the circuit of the CFLs they are designed to run from the peak voltage and not the RMS.  

You have IMO two choices (and I'm only suggesting this for driving CFLs!!!!):  

1) Increase you your input voltage. The turns ratio for your transformer is about 19:1 so to get closer to the 300 V peak you'd need about 17 VDC. You could use a boost convertor to provide this from your 12 V battery. 

2) Change your transformer, which may be harder. You'd need about a 9-0-9 volt transformer.  

Note.... big red flashing light ......you are setting the output voltage to 300 V RMS by implementing either of these solutions, so don't try to power anything other than the CFLs (or LED 230 VAC lights)
https://en.wikipedia.org/wiki/Root_mean_square