I am trying to work out the length contraction using the Lorentz transformations. Here is how I stated the problem:

Suppose a bar (whose proper length is $L$) is moving at speed $u$ (to the right) with respect to a reference frame $R$. Let $x\_2$ and $x\_1$ be the two coordinates of the rightmost and the leftmost points of the bar respectively. Measured in the moving frame $R'$, their difference is:
$$
x\_2'-x\_1'= \gamma(x\_2-ut\_2)-\gamma(x\_1-ut\_1) = \gamma(x\_2-x\_1)-\gamma u(t\_1-t\_2)
$$
But measuring a length means knowing both $x\_1$ and $x\_2$ at simultaneity in a frame. Therefore, $t\_2=t\_1$. It follows that:
$$
\Delta x'=\gamma \Delta x
$$
or
$$
L'=\gamma L
$$
Then
$$
\gamma = \frac{L'}{L}>1 \Rightarrow L'>L
$$
Which implies that length have dilated not contracted, as I was trying to solve. What is the problem with this reasoning?