You are wrong. Al-Jazeera is not state-run. To what extend the Qatari government tolerates it, I don't know, but al-Jazeera is not Qatari state TV.
"Al-Jazeera continues to broadcast because it reportedly receives $100 million a year from the government of Qatar. Without this subsidy it would be off the air, off the Internet and out of business."
Quote from Dorrance Smith, (April, '05) senior media adviser to Ambassador Paul Bremer.
As I have said previously, if Qatar wanted to, they could stop the hate America/pro-terrorist talk on AlJazeera, tomorrow.