iosiphonewifi
After updating my iPhone 3GS to iOS6, when I'm on WIFI the network connection appears terribly slow. Loading a webpage in Safari takes ages and opening the update tab in the App store seems to keep loading forever. When I turn off WIFI and try the same on a 3G network, it all is responding a lot faster.
I've checked the WIFI connection itself, and from my laptop that appears all fine. Before the update to iOS6, my iPhone was working fine on that same WIFI connection. What can I do?
The wifi icon (click here to see how you could have found about it on your own) is the top bar, right between the carrier and the clock, and it replaces 3G or E. It looks like this: 
Like Alvar said, best way to be sure is getting in Airplane mode and turning wifi on.
As for youtube not working while others do, it's probably blocked. This could be on the wifi or on the iPhone, if it's a company phone with a provisioning profiles blocking it or if it's jailbroken and got a firewall on it. Just check if youtube works on any other device on the same wifi and you'll know for sure where's the issue.
If youtube is just running slow, but it is running, it could also be your other heavy sites are cached. Without more info I can't think of more options for that issue.
Turns out the cause of this was...the baby monitor! If the monitor is on and close to the MacBook, it apparently interferes with the wifi. If I simply move it far enough away, that seems to resolve the interference.
Best Answer
Reset your network settings, to clear all remembered WIFI networks. This will force it to create a new ones. You will have manually connect to the WIFI network again, and if required enter its password again afterwards.
You can reset the network settings by selecting:
Settings -> General -> Reset -> Reset Network Settings
Also what can help is trying to clear the history of your Safari browser, you can do that by going to the Bookmarks -> History and then press the Clear button on the lower left hand corner of the screen.