Question: When watching the Sinquefield cup on <http://grandchesstour.org/>, it is possible to try different moves qnd see how the computer evaluates the position after your suggested move. 

Most moves will impair your position if it is not considered the best move by the computer. But for some moves (quite often) you instantly get an increased chance of winning, perhaps half a pawns advantage. I don't understand this: If a single move has a potential for significantly improving the players position, should this not already be reflected in the current standing before the move? 

In my mind, any move not considered the best move by the computer, should lead to an impaired position, while the best move should just at best keep the standing even, not actually improve the players position. Logically, if the computer knows this move will improve the players position, then shouldn't it already appear at the current position to improve the players position, saying that "with the best play, you are half a pawn up"? Alternatively, is this just a result of how the website is implemented at grandchesstour.org?

Answer: 
There is no inconsistency. When you make a move, the computer no longer has to search the previous position. It can now focuses all processing power into the new position, and thus the evaluation might change. The game tree could be very different.

This is just like how we play chess.
If there are 20 moves in a given position that means the engine is spending 95% of its time evaluating moves that are not the move that will be played. Switching from 5% of its analysis to 100% allows it to see much deeper much more quickly. This is especially true if the line chosen is a forcing line because the engine will be able to eliminate so many early moves.