Can linear regression or any other models be used to judge if Y increases as X increases? I have a database like:

Site    X    Y
S1      1    1.5
S1      1    1.3
S1      2    1.7
S1      1    1.1
S1      4    5.9
S2      3    4.0
S2      2    2.5
S2      4    9.1
S2      4    9.2
S2      1    2.3


I need to find if $Y$ increases as $X$ increases for every site. In other words, Bigger $X$ corresponds to bigger $Y$. 

I know linear regression might suit this problem. But please take a look at the following graph:



Figure 1 is not what I want because small $X$ corresponds to big $Y$. However, Figure 2 is what I want. When I use a linear regression model and RMSE as a measure, it cannot tell the difference between Figure 1 and Figure 2.

Another thing about my database is: $X$ are like levels, which are the same to all the sites. But the $Y\text{'s}$ of every sites are different. For example, for one site, $X=1$ and $Y=20$ means $20$ is a small value because it corresponds the lowest level of $X$. But for another site, $Y=15$ and $X=6$ means $15$ is the highest value because $X$ is the highest level.

So, my problem is: for every site, I need to use a linear model or any other algorithms to judge if $Y$ increases as $X$ does. Then, I need to use a measure to select some sites.
I don’t think linear model is a good idea for your problem, because it can capture only linear pattern. Furthermore, if I understand you correctly the results for different sites will not be comparable due to different scales/ slopes. 

Instead I suggest using the Spearman rank correlation coefficient.