Does a crosswind increase drag on forward motion?

Yes.

All winds are headwinds except for tailwinds. This is why riders in a bicycling peloton angle away from the crosswind behind the leaders.

Cars:

Engineers have various methods at their disposal to reduce the effects of crosswinds. things such as lower ride heights, lower overall height, stiffer suspensions, active steering and aerodynamics. (The egg shape of the Porsche 928 was the most visually obvious example of the latter).

Yes, because the body in motion has to steer towards the wind in order to keep proceeding straight.

In addition to what @zenvelo said, the steering has to be turned slightly to compensate for the sideways push and that increases friction/drag from the steering tires.

Good one! Let’s look at te vectors. Lets assumei the cross wind is perpendicular to the headwind vector. (Te car is hading down the road at 40 mph and the cross wind is 30 mph. If we allow the car to drift sideways with the cross wind we still are facing a 40 mph wind. No change. But unfortunately we have to keep the car moving straight. That makes the vector addition: (right angles) Headwind^2 + crosswind^2 = Net^2 . So in this example the net load would be the same resistance as if you were moving at 50mph.

Well done @LuckyGuy and everyone else. The principles become more readily appreciated when they are applied to an airplane, but folks make the mistake of believing the mass of the car counteracts sideways pressure like a fixed building. It only requires mounting that building on swiveling castors to appreciate that the structure would be driven merrily along the landscape by prevailing winds unless energy is brought to bear in resisting it.

What if we attached a sail to the car and drove at an angle to the wind? Boats can sail faster than the wind It is not clear from the article if your speed in direction of the road increases. I don’t fully understand how it works, but vector addition applies to forces, not velocities. Otherwise it would be impossible to sail faster than the wind.

@LostInParadise Boats can sail faster than the wind at times because the sail acts as a wing, it is not merely the strength of the wind pushing the sail.

I think you are referring to Bernoulli’s principle. I have seen articles saying that this is wrong , that the shape of the wing or sail makes no difference and that everything can be explained by the vector sum of forces.

@LostInParadise The car would not be moving faster down the road. (40 mph in my example.) However the forces due to wind loading would look like a 50 mph wind.
This was just a first approximation. Actually it is more complicated. Wind loads are (nearly) proportional to the square of the velocity. A typical road load for a passenger sedan is about 6 hp at 50 mph. Take it up to 100 mph and you can expect it to be 4x that 24 hp.
The EPA uses these figures to load the dynos that test vehicle emissions and performance. They are periodically tested and compared with measured coast down data.

What about the opposite of @stanleybmanly ‘s swivel caster car, one that had fixed front and back wheels and could only track in a straight line? Would a moderate crosswind make it deviate substantially from its trajectory? If not, then would that crosswind impede forward movement?