before attempt to turn, there was kinetic friction force only.
Where do you think you have kinetic friction? Kinetic friction is found when you have two surfaces sliding against each other. The tire-road interface is not sliding, so it is static friction, both before and during a turn.
During acceleration, the force of the tire on the road is rearward, so the reaction force of the road on the tire is forward. The only difference in the turn is that the frictional force is directed sideways instead.
If the forces between the tire and the road exceed the maximum static friction, then the tire will skid rather than roll.
The wikipedia article on rolling resistance points out that there are several mechanisms at work. While the scenario illustrated in your diagram (deformable tyre on a hard road) can be modelled as a torque opposing the rotation of the wheel (also friction at the axle), other scenarios (eg hard wheel on a deformable road) might not be.
The distinction between friction force, normal reaction and rolling resistance is an artificial one, not inherent within nature, so it is not sensible to be pedantic about the differences between them, unless the distinction is made in the question.
Like friction, rolling resistance is related by some empirical law to parameters such as the normal reaction between the surfaces, the diameter of the wheel, tyre pressure, or the linear velocity of the vehicle. The law may contain several coefficients, and relates to a specific combination of materials. The wikipedia article provides examples.
Unless there is some particular reason to do otherwise, I suggest that the simplest solution is to model all forms of rolling resistance (whatever the mechanism) as a single force which opposes rolling motion.
In answer to your final question, there could be different combinations of rolling resistance and static/kinetic friction on each tyre of a moving vehicle, depending on the circumstances.
In the ideal case (no rolling resistance), constant velocity requires no force, so there is no static friction. If rolling resistance is not zero then there must be a driving force and therefore some static friction acting on all wheels to keep them moving at constant speed. If the vehicle is accelerating/decelerating then static friction is required to speed them up. If the vehicle is braking there may be some kinetic (sliding) friction on the braking wheels.
If all the wheels are the same, and bear the same load, then the rolling resistance is assumed to be the same on each, whether they are driving or braking or neither. Some wheels may bear a greater load, eg because the heavy engine is closer to them, or the car is accelerating or braking heavily. For those wheels rolling resistance will be higher.
Best Answer
Static if there is no relative motion between the ground and the tyres at the point of contact.
If it was a block then as there was relative movement between the block and the ground then it would be kinetic friction.
You need to produce a centripetal acceleration and so need to provide a force towards the centre of the circular trajectory.
Although the cycle is moving forward if there is no slipping at the point of contact between the ground and the tyres (and also there are no other frictional forces eg air resistance) no tangential force should be needed to maintain a constant speed.
If it was a block you would need a tangential force to maintain a constant speed and a radial force to make the block move in a circle so the net foce would be at some angle between the radial and the tangential forces.
You only need the car engine to do work if there are frictional etc forces acting on the car and you want it to maintain a constant speed. In a lot of Physics problems the frictional forces are assumed to be zero.