"curl is an invariant quantity under a Lorentz transformation" is not correct. It depends on curl of what. More fully, the curl should be thought of as part of a 4-vector type of differential operation which can be written
$$
\frac{\partial A^a}{\partial x^b} - \frac{\partial A^b}{\partial x^a}
$$
where $A^a$ is a 4-vector. Once you know this you can also learn the effect of a Lorentz transformation on it, but I am guessing you have not learned this area to this extent yet so I won't go into it.
In the case of electromagnetic fields, both the electric and magnetic fields transform from one reference frame to another: they are not invariant. In fact a nice way to find both the electric and the magnetic field around a charge moving at constant velocity is to start from the purely electric field of a stationary charge, and then change reference frame. But of course to use this method you need to already know how the fields change. Google will soon lead you to more information on that if you wish to explore further.