Permit me to amplify what "PleaseStop." has said. And I'll repeat it even once more, at the risk of sounding like an echo chamber:
Gravity is NOT a force in GR.
It is a 'pseudoforce' -- it only seems to be a force because of a poor choice of one's frame of reference. In an accelerating frame, there is always a pseudoforce that is needed to make F=ma work. A familiar example is the so-called "centrifugal force" in a uniformly rotating frame.
In Newton's model of gravity, space and time are immutable, space is 3-dimensional and Euclidean (that is, its 'metric,' the rule that tells how to determine squared distance from coordinates, works according to Pythagoras' Theorem), time is an independent variable, both are unaffected by motion or mass; they are the 'background canvas' on which all physical phenomena are painted. Galilean relativity rules the mathematics of motion, and gravity is a force that acts mysteriously and instantaneously at a distance, between all massive objects in the universe. Philosophically, Newton was sorely troubled by this picture of gravity, but, as it explained observable phenomena so well, he just left it there. "Hypotheses non fingo." -- I make no hypotheses [about a causal mechanism for gravity]. (This is the conceptual problem you highlight in your last statement/question, under Additional Details.)
By the mid-to-late 19th century, this model had come into seemingly irreconcilable conflict with the lately well-established laws of electromagnetic (EM) phenomena. Some thought that the laws of mechanics would have to change; others thought it was the laws of EM that needed revision. Einstein found a third way -- change the description of the canvas itself!
So in Special Relativity (SR), space and time are a unified 4-dimensional entity, with a Minkowskian metric (same as Euclidean, only with signs on the 3 space-squared parts that are opposite the sign on the time-squared part). The "exchange rate" between space and time is c, the 'speed of light in vacuum,' which is constant.
[My own 2 cents -- c should properly be called, "the characteristic spacetime speed," because it is really a property, not of light, but of spacetime itself. Any massless field will travel at speed c.]
Once he had this all worked out (1905), he realized right away that gravity in the Newtonian formulation was untenable in SR. So after 10 more years, he came up with General Relativity (GR), in which, as "PleaseStop." says:
"Matter tells space how to curve; space tells matter how to move."
[in which the term, "space" is now understood to include time.]
-- a description coined by John Archibald Wheeler, who also coined the term, "black hole."
In Einstein's field equations for GR, the source term is not just mass, but also energy and "stress." So, yes, in GR, light, which is massless but has energy, does participate in producing 'gravity,' that is, it contributes in a prescribed way to the curvature of spacetime.
And Einstein's picture of gravity does away with action-at-a-distance, making physics local once again, because it has mass-energy-stress creating a local curvature of spacetime, which then propagates (at speed c) to neighboring regions of spacetime, and so on ... and Newtonian gravity is a pseudoforce in this picture; due to working in a noninertial frame. Inertial (free-falling) frames near Earth's surface are accelerating wrt that surface at 1g, downward. So standing 'still' on the floor, you are actually accelerating at 1g away from Earth's center, pushed up by the floor. The fact that neighboring inertial frames are accelerating slightly wrt to each other, at a rate that depends on their separation, and on their direction of separation, is simply one manifestation of the curvature of spacetime. GR describes the tidal (gradient of) force.
EDIT:
Thanks, PleaseStop. Bravo right back to ya.
I had to weigh in when I saw your emphatic denial that gravity is a force, which I've tried to impart many times in this forum in questions about gravity.
That said, I might not have been quite so hard on gintable; he makes a number of important points. While a single photon won't act exactly like a particle of mass hν/c^2 & momentum hν/c (for one thing, it travels a light-like, not a time-like geodesic of spacetime), yet if we had a massless, perfectly reflective box containing a photon gas, I believe it would behave just like a body of mass equivalent to the energy of its photons, for purposes of gravitation, as well as in response to any mechanical forces. In any case, it would take a phenomenal concentration of photons to produce any noticeable gravitational effect. I'm guessing it would take something like a supernova event to form such a concentration, and that to treat such an event in GR would require considering the effect of those photons on the gravitational field. I believe that would come close to an experimental verification that light participates in gravitation.