Is angular momentum conserved in case of a vehicle turning?

Suppose that a car is driving in a circle with constant and speed. In this case its angular momentum about the centre of the circle is conserved because the friction force that keeps it in its track is radial.

If the car drives on a straight line with constant speed then angular momentum is also conserved, with respect to any axis.

In general the car should drive such that its acceleration is radial and falls off as 1/r. Sounds familiar? Indeed, this defines a Kepler orbit.

I neglected relativistic effects and the effects of the curvature and rotation of the Earth.

Your premise for applying the law of conservation of angular momentum is wrong. The law can be applied if you find an axis about which the net torque on a given system is zero.

In the case of a car taking a turn, the axis location changes from infinity to a closer point. The center of curvature of the car's path is not a valid axis as it changes with time.

If you are a little more specific regarding the motion of the car, it might be possible to locate such an axis, but if you already know the motion of the car, such an axis is of little practical use.

Angular momentum is given by $mvr$. Angular momentum is a vector quantity.

In this case angular momentum is not conserved in this case as direction of velocity is changing.

the car turn by an outside force (friction in the wheels) which slows down one part more than another. if you were to turn it batman style- by grappling hook to a lamppost I'd imagine that would be the case (not sure though, not an actual physicist)