Suppose an object measures $L$ in moving frame $S'$. This is measured at the same time so $\Delta t'=0$ :

$$\Delta x = \gamma(\Delta x'+v\Delta t') = \gamma \Delta x' = \gamma L$$
Since $\gamma \gt 1$, the same object is dilated in the rest frame $S$. This is wrong as we know that the object must be contracted. What am I doing wrong?

Suppose an object measures $L$ in moving frame $S'$. This is measured at the same time so $\Delta t'=0$ :

$$\Delta x = \gamma(\Delta x'+v\Delta t') = \gamma \Delta x' = \gamma L$$
Since $\gamma \gt 1$, the same object is dilated in the rest frame $S$.
This is clearly wrong as we know that the object is actually contracted. What am I doing wrong?