Has relativity of simultaneity been directly observed? I know that thought experiment about trains when a flash of light in the middle reaches the both end simultaneously for a passenger but different times for the bystander.
So were there (non-thought) experiments that directly checked this?
 A: The most direct experiment confirming this is the Kennedy-Thorndike experiment. It used two different length arms so modifying the Michelson-Morley experiment that had shown that no local "aether" explained observations calling for an explanation that Einstein provided. 
P.S. If you want seek more technical explanation of the methods, it is readily found on the Web using those names.
A: I think the answer to this depends a lot on your definition of "directly." Relativity of simultaneity is built into the Lorentz transformation, and Lorentz invariance is one of the most precisely tested physical theories in all of history. Essentially you're asking for an experiment that verifies one element of the matrix involved in the Lorentz transformation, but every element of the matrix is present in all cases. I would consider the Sagnac effect to be a fairly direct test, and the Sagnac effect was one of the effects observed in the Hafele-Keating experiment, as well as many other, earlier tests of relativity. Every time you fly on a commercial jet, you're benefiting from a  ring laser gyro, which works based on the Sagnac effect.
A: 
[...] that thought experiment about trains when a flash of light in the middle reaches the both end[s] simultaneously for a passenger but different times for the bystander. 

That's not really a great/correct description of (a variant of) the well-known thought experiment, closely related to Einstein's (final and enduring) definition of how to measure "simultaneity". 
Instead: if some (third) passenger in the middle between two given passengers on a train (e.g. the two ends of the train) stated a signal (flash of light) then the corresponding indications of the two given passengers/ends receiving this signal are simultaneous;
and if some railroad tie in the middle between two given bystanders stated a signal (flash of light) then the corresponding indications of the two given bystanders receiving this signal are simultaneous as well.
It is important to understand that the railroad tie is not identified as middle between the two given passengers, and that the mentioned third passenger is not identified as middle between the two given bystanders; not even "at the moment they pass each other". The railroad tie always moves relative to the ("moving") train with all its passengers; and all passengers (in particular the third one, in the middle between the given two passengers) always move relative to the bystanders.

So were there (non-thought) experiments that directly checked this?

There are no meaningful experiments without underlying thought experiments.
There cannot be any measurement values obtained from observational data without first deciding on (and then sticking to) some particular method of measurement.
To paraphrase Niels Bohr: We must be able to tell our friends what we have done (or at least intended to do), so they'll also listen to what we have found.
And of course there is no experimental check of definitions (such as Einstein's definition mentioned above), or of terminology, or of "rules to play by";
there are ("only") experimental checks of hypotheses expressed in terms of definitions and terminology and "rules" (which one has decided to use, and which ought to be stuck to), such as:
whether bystander Alice took a sigh simultaneously to bystander Bob picking his nose, or not; or
whether the (third) passenger's name in the middle between the two given passengers Reese and Dewey was named "Malcolm", or not; or
whether all passengers moved equally (at the same velocity) relative to the bystanders, or not; etc.
In short: asking whether experiments "checked this"
is absurd without thought-experimental definition of what "this" is supposed to be. 
A: It is logically wrong to state that the experiments mentioned in the other answers prove the relativity of simultaneity, since there are alternative theories to special relativity in agreement with all experiments to date in which absolute simultaneity is preserved. In other words no experiment has allowed to observe the relativity of simultaneity.
See the Mansouri and Sexl paper "A Test Theory of Special Relativity", I'm quoting the abstract:

An ether theory is constructed that maintains absolute simultaneity and is kinematically equivalent to special relativity

