Is the intuition of “absolute simultaneity” necessarily wrong? I read about Einstein’s theories of relativity with all the interesting stuff like time dilation, mass increase and whatnot. And also about the relativity or conventionality of simultaneity. The latter I found the most interesting.  I now know that relative simultaneity is at the least assumed to be true. And that the experimentally well verified theories of relativity have relativity of simultaneity as an axiom.
But was it proven that there is no absolute simultaneity ? Is there any practical experiment to distinguish between relative and absolute simultaneity ? Has it been done already ?
My intuition tells me that that proving the truth of relative simultaneity compared to absolute simultaneity is impossible, except if time-travel into the past was possible. However my intuition also tells me that the absence of absolute simultaneity is an inconsistent idea, which would contradict the very theories of relativity. 
So I’m very curious what answer the people of physics, who should know this stuff better than me, can provide to my question.
 A: The truth of the Lorentz transformation as an accurate description of the co-ordinate transformation between relatively uniformly moving observers needfully implies relativity of simultaneity. Contrapositively, the Lorentz transformation cannot be sound if simulteneity is not relative. So, in the sense that the soundness of the Lorentz transformation has a great deal of experimental support, there is also strong experimental disproof of the notion of absolute simultaneity. The most in-one's-face results (IMO) are the measurement of varying lifetimes of metastable particles depending on their speed relative to the laboratory; the lengthening of lifetimes is precisely in keeping with the time dilation factor calculated from the Lorentz transformation. See also the Rossi-Hall and Frisch-Smith experiments.

My intuition tells me that that proving the truth of relative simultaneity compared to absolute simultaneity is impossible, except if time-travel into the past was possible.

Actually it's not anything like as wildly arbitrary as this, and this is one of the most astonishing things about the Lorentz transformation[1]. Whilst the simultaneity of events depends on the observer, the order of causally related events is not relative. So if a cause comes before an effect in one frame, it does so in all frames. Indeed, this is precisely where the notion of no signalling faster than lightspeed comes from: the Lorentz transformation is such that this order preservation is sound as long as we postulate that cause-effect relationships cannot propagate faster than light. In more technical language: the order of timelike-separated events cannot change under the Lorentz transformations, whereas that of spacelike separated events can. So we postulate that cause and effect must define timelike separation: then we recover our altogether everyday experience that causes always come before effects and relativity does not mess with this everyday experience in any way. Notwithstanding time dilation, twin paradoxes and all the rest of it, it's actually quite remarkable how sound our everyday physical intuition remains in a relativistic treatment.
[1]. In this answer I talk wholly about proper, orthochronous Lorentz transformations (i.e. the identity-connected component of the Lorentz group). These are the transformations comprising rotations, boosts and combinations of these two: to wit: the transformations that can be realized by relative motion. The full Lorentz group includes time reversal and reflexion operators, but these cannot be realized by relative motion.
A: The relativity of simultaneity is not an axiom, the axiom is that the light velocity is the same in every frame of coordinates. A spot of light travels at the same velocity, c with respect to you, and at the same velocity $c$ with respect to a traveler traveling with respect to you at an arbitrary velocity. 
So, assume that you send two spots of light to two opposite points, A and B, at the same distance from you, see the figure. According to your clock they reach the points A and B simultaneously. 
But consider a traveler in the direction from you to A. She will find, according to her clock, that the spot of light reaches the point A before the other spot reaches B. That happens, in simple words, because the light moves with respect to her at the same velocity, c, while the target A "comes" toward the traveler while the target B recedes from the traveler. So, the light spot will hit the point A before the other spot hits B.
A traveler in the direction from you to B will find opposite order of hitting the points,ly from symmetrical reasons. 
As to experiments of this type, they weren't done so far, but the theory of relativity was tested in different other ways.

A: Yes, in modern physics the idea of absolute simultaneity is definitely shown to be incorrect.
That said, it is possible that some non-physicists use the term absolute simultaneity in a loose way to convey the idea of a universal 'now', which is quite a different question.
It is now well established, by hundreds of experiments, that the speed of light is the same for every inertial observer, and from that it is easy to show that simultaneity is relative. There is a very simple thought experiment you can follow to convince yourself of it.
Suppose I flash a light that heads off in all directions around me. Suppose that there are two detectors equidistant from me, one to the east and one to the west. Let us say they are each 299,792,458 metres away, which is the distance light travels in a second. If I flash the light at exactly 12:00:00, the two detection events will occur at exactly 12:00:01, so I would say they are simultaneous, because the time required for the light to reach one detector is exactly the same as the time required for it to reach the other.
Now imagine that you are walking past me, heading east at a meter per second, at exactly the moment I flash the light. Your watch is synchronised with mine, so we both agree the light was flashed at 12:00:00. As before, the two detection events occur equidistant from me at 12:00:01. However, in your frame, the eastern detector must register the light earlier than the western detector, because in your frame the light has had to travel a metre less in the eastern direction, and a metre more in the western direction. In your frame, therefore, the eastern even occurred fractionally before 12:00:01 while the western event occurred fractionally after 12:00:01, so they were not simultaneous.
Crucially, however, you and I do not disagree about where and when in spacetime the two events occurred- we simply use different coordinate systems to label them. In my coordinate system, the two events have the same t coordinate, while in your coordinate system they don't. There is an objective reality, or an absolute reality if you prefer, about the events having occurred at two specific points in spacetime- they fact that you and I label them differently is in many respects immaterial.
If you are contemplating the philosophy of time, you should bear in mind that simultaneity has a very limited meaning in physics. You might have heard of the Andromeda paradox, which is really just an extension of our thought experiment to massive distances. If you are walking past me, then for you the time on a distance galaxy that is simultaneous with the time on your watch might be several years before or after the time on the galaxy that is simultaneous with the time on my watch, the difference depending on how far away the galaxy is, and the direction in which you are walking.
You might find it helpful to consider the paradox in reverse. Suppose on the distance galaxy there are three people standing together, stationary relative to you, their time synchronised with yours. Now suppose one of the people starts walking away from Earth and one towards it. For the one walking towards you, the simultaneous time on Earth shoots years into the future, while for the one walking away from you, the simultaneous time on earth leaps years back in the past. What difference does that make to the objective reality of 'now' here on Earth? Absolutely none.
A: Assume two space-stations at rest with respect to one another - therefore their clock-beat is the same. In one station works the experimenter Alice, and in the other station, the experimenter Bob. Consider that at the time $t_0$ by the clock of these stations, each experimenter performs some experiment.
Assume now that the experimenter Charlie is sent to collect the results of the two experiments. Charlie's station is located on the same line as Alice's and Bob's stations, beyond Bob's station:
Charlie_____Bob___________Alice
Charlie decides to start his flight so as to reach Bob's site right in time when Bob does his experiment; immediately after that, Charlie intends to continue to Alice. According to the clock of the ship, Alice performs her experiment before Bob, s.t. when Charlie would be at Alice's station, her result would be ready.
However, after Charlie's ship lands on Bob's station, the beat of the clock of the ship becomes identical to the beat of the clocks of Alice's and Bob's stations. Therefore, Alice appears as performing her measurement again, simultaneously with Bob.
Thus, from the point of view of Charlie, Alice performs her experiment twice. But this is false, Alice performs her experiment only once.
Thus, when passing from one inertial frame to another, the simultaneity changes, and may lead to contradictory descriptions of events.
Best regards!  
A: The notion of universal, absolute simultaneity is broken by the existence of singularities:

a condition in which gravity is so intense that spacetime itself
breaks down catastrophically. As such, a singularity is by definition
no longer part of the regular spacetime and cannot be determined by
"where" or "when".

from Wikipedia: Gravitational singularity

Whether absolute simultaneity is the rule for regular spacetime appears to be elusive to prove, e.g. Thomas Ryckman's 2010 review of Einstein, Relativity and Absolute Simultaneity.
Nevertheless, the OP asks whether it has been "proven that there is no absolute simultaneity?", which is a different matter.
Absolute simultaneity would be beyond observability, and

An outstanding characteristic of modern physics is the application of
the principle that only that which is observable is significant.

from Nature 1937 Science and the Unobservable.
So defining absolute simultaneity in terms of physics is going to meet challenges, despite conceptualisation in satellite GPS synchronisation.
On the plus side, closed timelike curves that violate causality are theoretical and do not amount to proof or disproof.
e.g. Closed timelike curves and causality violation (2010)
A: 
My intuition tells me that that proving the truth of relative simultaneity compared to absolute simultaneity is impossible, except if time-travel into the past was possible.

This is false. You could set up an experiment where you make two flashlights flicker simultaneously in your frame, significantly far apart in space.
Then you could have an observer $A$ in relative motion at speed $v$ with resepect to you detect the light from both flashlights. After $A$ subtracts the light travel time, they will observe a non-zero time-interval between the events.
For a simpler experiment, you could have a third observer $B$ traveling with the same speed $v$ but in the opposite direction. One flashlight is red, and the other is blue. Then $A$ and $B$ will disagree about the order that the events "red flicker" and "blue flicker" happened. The time interval between the flickers could even be years depending on $v$, so it would be impossible to account for this using the travel delay of light. If unsatisfied, you could again subtract the light travel times and still see a discrepancy in the order
