None of the DS web browsers support flash video like Youtube. The rest of the web is mostly accessible depending on the browser you use.
Okiwi and Bunjalloo and the "links -g" browser of DSLinux all operate on the DS hardware itself and its 4 MB of RAM. DSLinux uses any external slot-2 RAM it can find, which is necessary if you want to use the graphical version of Links. All of them are slow, though the official Opera browser is the most accurate.
Honestly, while the DS web browsers worked for me in highschool, I abandoned them the moment I got an iPhone. It still renders pages slowly, but it's so much better than the DS browsers that it makes me wonder how I put up with it for all those years.
EDIT: use Costello's DSHobro for the difficult pages, as demo'd at
this Youtube link, but don't expect Youtube on the DS. Netflix on the 3DS is the closest to that.