In short, no. Websites hosted by reliable, well known companies like Google/Microsoft/Apple/etc generally are extremely secure, and have administrators constantly maintaining them. Although it's not unheard of for them to be hacked, such hacks usually only result in down-time, as the web-admins will take down compromised servers immediately upon discovering the intrusion (usually with the assistence of Intrusion-Detection software, like Tripwire). Spreading malware through big websites like those is very rare.
Websites that may be compromised and spread malware for a long period of time/forever are usually smaller, one-man run websites, like fansites. Often such sites are maintained by people lacking in advanced server security knowledge, and hence an intruder may compromise the site without the owners knowledge.
People that believe their computer was infected by big-name websites like Youtube usually received the virus from another website, and the malware merely impacted their use of Youtube, resulting in them believe the problem to have originated from Youtube/Apple.com/etc.
