What specific R9 270X GPU are you using? If you're 600W power supply has enough connectors for the PCIE 6/8-pin required by your GPU, then, it is an indication that your PSU can provide enough power.
What is the exact model of your 600W PSU? I'd also check the +12V combined power/amps the PSU can provide (since it is going to be definitely lower than the "marketing wattage" of 600W).