If the moving object bends space-time

Does a moving object bend space-time with increasing speed?

I suppose you are wondering if a moving object is spacetime different curves as a stationary one as its speed increases.

Strictly speaking, no: it is the same spacetime geometry in both cases. The spaceship distorts spacetime in both directions, and we would just talk about it in a different frame. Because of this difference in frames, the gravitational field is, in a sense, different even though the geometry is the same.

According to the special theory of relativity, if this spaceship gains mass as a factor y as it approaches c, its gravitational field should also increase in strength.

That is not completely right. First, the spaceship is not gaining mass. It is true, this set γm is sometimes called called relativistic mass but this term is redundant with energy, poor in its intended purpose of maintaining a superficial resemblance to Newtonian mechanics, and devalued in physics. In the special theory of relativity, the mass is invariable: (mc2) 2 = E.2- (pc) 2 is the same in all frames of inertia.

That is a good thing, because the "gravitational charge" is not mass, but energy. But it's not a simple proportional increase when we look at the spaceship's gravitational field in a frame where it has a lot of energy.

This should come as no surprise if you know some electromagnetism. A moving charge creates an electromagnetic field that has both electrical and magnetic parts because the movement of the charge, i.e. the current, is important. The electric field is intensified in directions perpendicular to the direction of motion, which we can think of as the initially spherically symmetrical field lines that are Lorentz-contracted, thereby "squeezing" them close together in perpendicular directions.

The gravitational analog of electric current would be momentum, but since gravity is spin-2, the voltage is relevant to the bending of space-time in addition to the energy density and momentum density. You can see this in the stress energy tensor. So the gravitational field is more complicated, but it has an analogous behavior to be amplified perpendicular to the direction of movement, although its quantitative behavior is different.

At the limit of the speed of light, the electromagnetic field of an electric charge changes into an impulsive plane wave, and the gravitational field of a point mass behaves analogously and changes into a vacuum impulsive gravitational wave (pp-).