Elaborating a bit more on what izua has said, if you check your datasheet, you'll see a spec called "total voltage drop" or something similar. This is the voltage drop between your input and your output and it's not something you can make better by any means other than increasing your input voltage. If you want a smaller voltage drop, look into using MOSFET-based drivers (MOSFETs typically have much smaller drops and they have some other benefits as well).
Looking at the datasheet for your driver, it would appear that the typical high output voltage is VIN - 1.4 and the typical low output voltage is 1.2, so the voltage across your motor is typically somewhere around 2.6 V lower than VIN when the motor is pulling 1 A. What this means is that when you try to power your L293B using a 5 V power source, your motor is only seeing perhaps 2.4 V across it (maybe more, maybe less, depending on current).