The curvature of spacetime depends on the metric tensor. A tensor is an object that does not depend on the coordinate system, although its components are different in different coordinate systems. This is similar to a vector: if you change the coordinate system, the components of a vector would change, but the vector itself would not (except of course for the radius vector).
For a non-accelerating object that is relativistic in your frame, the object is stationary in its proper frame. The metric tensor in this frame is defined by the invariant mass (a.k.a. rest mass). Now, in your frame, the components of this tensor will be different, but the tensor itself, as an object defining the spacetime curvature, would be the same.
This is different for light, because it is impossible to define an inertial frame, in which light is stationary. It gets even more interesting from here. The photon does not generate a static gravitational field. The gravitational field of the photon is a gravitational wave perpendicular to the direction of the photon and moving with the speed of light.
The fact that both the photon and its gravitational wave move with the speed of light works out to a remarkable result. From the viewpoint of the observer affected by this gravitational wave, the wave is not associated with the photon, but with the spacetime events of the photon emission and absorption, where the former is seen as an attraction while the latter as a repulsion:
The gravitational field of photons
The gravitational field of a laser pulse
The result that the gravitational wave of the photon is associated with the emitter is remarkable, because photons do not experience time and therefore cannot change in flight (e.g. emit gravitons). This is similar to the fact that neutrino oscillations are the evidence that neutrinos do not move with the speed of light. However, a full understanding of the gravity of light requires developing a theory of quantum gravity.
You can often hear an argument that light trapped in a massless box with ideal internal reflection has invariant (rest) mass and therefore creates the same gravitational field as a massive object of the same mass. This is incorrect. The gravitational field would not be the same.