Why didn't Jesus give proof to the priests? On several occasions the High Jewish priests asked Jesus, that if He is the son of God, then demonstrate by doing X or Y. 

I assume in all of these cases, he could have fulfilled the requests to show who He was - so why not just give them some proof?  

Why did Jesus have to beat around the bush ?

Why not prove Himself for exponentially more followers?
Jesus did give them proof, on numerous occasions. He healed the sick, turned water into wine, and raised the dead. He showed how his coming fulfilled prophecy. But they still didn't believe.

Consider the story of the blind man Jesus healed in John 9. When he was brought to the Pharisees, did they say, "Zounds, Jesus restored the sight of a blind man! He must be a great prophet if not God himself!" No, they didn't. They had the man right there, giving his eye-witness (no pun intended) testimony. And what was their response? They had his parents brought in and questioned if he had really been blind. When they said he was, they demanded that the man explain how Jesus did it. Of course he didn't know. They pressed and pressed, and when he wouldn't recant his story, they had him excommunicated.

So when they asked for yet another sign, Jesus told them no, he had already given them all the proof they needed. If Jesus had come to some meeting of the priests and performed a miracle right in front of them, I'm sure they would have claimed it was an illusion or a magic trick.

It's just like some of the demands people make today. Every now and then an atheist will say, "If there really is a God, let him strike me down with lightning right now." Why doesn't God respond by sending lighting? Because: suppose that an atheist said that, and at that very moment a bolt of lightning tore through the roof and knocked him to the ground. Would all the atheists say, "Wow, we were wrong. God did indeed prove his existence"? I really really doubt it. Much more likely they would say, "Wow, what an astounding coincidence! Just as Bob made that comment about God striking him with lightning, he actually was struck by lightning! You know it would be funny if it wasn't that poor Bob was so badly hurt ..." Right? Does anyone seriously think otherwise?

At some point you have to say, he's done all these miracles, he's given you the Bible, you can believe it or not. In Luke 16:31, Jesus says, "If they do not hear Moses and the prophets, neither will they be persuaded though one rise from the dead." I used to wonder what Jesus meant by this. For of course, if someone really came back from the dead, everyone would have to believe. How could they not? Then stories of near-death experiences came out, people who claimed to have died during surgery or in accidents and then been revived, and who talked about seeing a "being of light" and a heavenly -- or sometimes hellish -- place. Did this end the controversy, and everyone who heard became Christians? No. They said these people were making up stories or hallucinating. Maybe some people became convinced that there is life after death from such testimony, but I suspect it's few. I'm a Christian, I believed in life after death before I heard these stories, and they don't particularly convince me of anything. I don't know what to make of them.

If someone doesn't want to believe, no amount of evidence will convince him. God's not going to waste his time giving a 127th proof to someone who wasn't convinced by the first 126.