I’m having similar issues with two Macbook Air’s - both running 10.9.4.

One was running Avast 9.0 and 99% of the web traffic was blocked - only Google would work (Avast.com was blocked too). Turning off Web Shield and everything worked.

Another was running Avast 8 and everything worked well. To troubleshoot I upgraded to 9.0 and then my internet was blocked too. Again, turning off Web Shield allowed everything to work again.

There is something with 9.0 and 10.9.4.