I would say "put your hand up" as to me it sounds the most natural. "Put up your hand" works as well. "Lift your hand up" is bit strange but you could say it. I feel it implies that someone's hand is already up and you want them to lift it higher. You wouldn't say the third option ("turn up").

I’ve heard put your hand up but it’s mostly raise your hand:)

"Put your hand up" is the standard expression in the UK, and also probably in other countries which tend towards the British model. "Put up your hand" would have the same meaning. "Raise your hand" is also correct, but less common in the UK. It has a rather formal sound to it. "Lift up your hand" is a little strange. In context, it would probably be understood, but it isn't quite right. "Turn up your hand" is makes no sense at all. [The only context in which 'turn up' can mean 'raise' is when you are talking about the volume on a sound device. The image here is of turning a dial or knob clockwise to increase the volume - like on an old-fashioned radio.]

There appears to be a difference between British and American English here. In the UK 'put your hand up' would be the most likely expression in this context, while the American answerers would not use this expression and find it a bit strange.

In that context, raise your hand is really the only one that's appropriate (in American English, at least). It's an expression that's used fairly consistently, and I can't think of any alternatives that wouldn't sound awkward to a native speaker. Though it's not quite the same, there is a similar expression, using "show of hands." Usually, it's used when you're looking for a general count: "Can I have a show of hands? How many of you did the reading?" "Could I get a show of hands? Who wants to meet on Friday?" "Give me a show of hands if you agree with me." Hope that helps!