My impression was that the curl of a vector field measures how fast a vector field turns along a closed curve around that point. Consider the vector fields $\vec{V}_1=-y\hat{i}+x\hat{j}$ and $\vec{V}_2=2\vec{V}_1$. If we consider the closed line integral (or circulation) of $\vec{V}_1$ and $\vec{V}_2$ along the circle $x^2+y^2=1$, it is clear that the circulation is $2\pi$ in the first case $4\pi$ in the second case. Also, the curl of $\vec V_2$ is twice that of $\vec V_1$.
This suggests to me that the field $\vec V_2$ turns twice as fast as $\vec V_1$. But if we plot both fields on Mathematica and also draw the contour $x^2+y^2=1$, the fields seem to turn equally along the curve i.e. both the fields turn by $\pi$ when we complete half the circle and $2\pi$ when we complete the full circle, exactly once. Is my intuition of line integral or curl wrong?
I suspect you have a fundamental (but entirely understandable) misunderstanding of what "how fast a vector field turns" actually refers to.
The following is a standard intuitive visualisation of what curl measures:
If your vector field is something like winds, or a water current, and your circle is an actual physical object that the fluid flow can turn freely but cannot move, the circle will rotate according to the integral of the curl on the region inside it. Which is to say, according to the integral of the component of the vector field that is tangent to the circle.
It is more generalisable to imagine the circle as a fixed rail that you can fill with beads, and the beads can glide freely along the rail. This interpretation is more correct for non-circle paths.
So the vector field itself doesn't have to go in a circle. A positive curl at a point does mean that the vector field has a tendency point counterclockwise around the point. But it doesn't have to actually point counterclockwise everywhere. It's enough that it points more counterclockwise than clockwise. This tallying is made more rigorous by integrating the tangent component of the vector field along the circle.
Consider my example above: $(x+3)\hat j$. It points upwards everywhere on the unit circle. So it points counterclockwise on the right half and clockwise on the left half. But the vectors on the right half are larger than those on the left. So there is net positive counterclockwise pointing, which aligns well with this vector field having positive curl.
A larger curl (such as for your $\vec V_2$ compared to $\vec V_1$) means that the circle (or the beads) turns faster as it lies in the in the flow. It means that the net counterclockwise pointing (as calculated by the integral) is larger.