12V motor running with 24V L293D H-bridge
The datasheet for your selected H-Bridge indicates a maximum amperage of 600mA per channel. It seems like this should be enough for your 12v motor. (whats the model number of the motor?) 

So this could mean that your power supply is the problem. You are thinking about raising voltage:


But unless I misunderstand, what you need to do is this:


Unless of course one of your motors draws more than 600mA then you need a different H-Bridge

References:
battery images credit goes to http://www.zbattery.com/.
Texas Instruments datasheet http://www.ti.com/lit/ds/symlink/l293d.pdf