You say that only web access is affected, not mail? This makes me wonder if something's going on at the ISP. It's hard to tell exactly what from here, but you should not need to restart the whole machine. Try right-clicking on the wireless connection icon in "Network Connections" and say "Repair".
What to do next depends on your level of knowledge. If I suggest temporarily changing the connection's DNS setting to use Google's DNS (8.8.8.8), as a test, do you know what I mean by that?