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At the time the Big Bang happened the matter had enormous density. According the GR (I may be wrong here) such density dilates time.
If so, could it be that the time periods just after Big Bang which are usually considered happening in small part of a second (such as the Planck epoch), in reaity took billons of year (or may be, infinity) but due to time dilation appear to us as spanning only microscopic parts of a second? Could it be that the age of the universe is dramatically underestimated?
there are certain laws that established by experiment (e.g. time diliation in presence of high density).
There is no such law.
So what the basis for claiming this did not happen in early universe as well?
The fact there is no such law, for one.
Ok, let's step back a bit and take the simpler examples of time dilation. In special relativity, the ur-example of time dilation is between inertial frames, which correspond to a family of comoving intertial observers. So in making a statement involving the time dilation factor $\gamma$ such as $$\frac{\mathrm{d}t}{\mathrm{d}\tau} = \frac{1}{\sqrt{1-v^2}} = \gamma\text{,}$$ we're comparing one inertial observer (with time $t$) with some other, not necessarily inertial, observer with time $\tau$. This is also generalizable to comparing arbitrarily-moving observers.
For gravitational time dilation, the ur-example is a comparison of the passage of time for two different observers stationary at two different elevations in a gravitational field. The moral is just the same: any talk of time dilation is at least indirectly referring to a comparison between different clocks, and thus different observers.
It basically boils down to: time dilation of what relative to what? Once you understand this, you will see how your question is actually reducible to the previously linked one.
It seems to me that after performing this the Big Bang will be at infinite past (am I wrong?) or in other words, there would be no big bang at all. I understand that this amounts to "interpretation" by I still do not see why this time scale is worse than the stand one
You can always formally define a time $t'$ such that the big bang occurs at $t'\to-\infty$, or some such thing, e.g., by $t' = t_0\log(t/t_0)$, where $t$ is the usual cosmological time and $t_0$ is some arbitrary time scale. But doing this is merely a relabeling of coordinates, and has no physical significance. In particular, it doesn't change the length of the worldline of any observer, which is the duration measured by a clock along that worldline, aka proper time.
And in the usual Big Bang cosmology (though not all FLRW solutions), the proper time of any observer, however idealized, is finite in the past. Relabeling to infinite time is no different then taking the usual Euclidean space in Cartesian coordinates, $(x,y,z)$, defining a new coordinate $\xi = \log x$, and then saying that the origin is now infinitely far away, being at $\xi\to-\infty$. You can define the coordinate, but it doesn't change actual distances.
If I remember Stephen Hawking alludes to some sort of "quickening of time" near the big bang in his book A Brief History of Time (though I read it 20 years ago, so don't hold me to the exact details). He makes an argument that if time is measured by the number of collisions in a given volume then the number of collisions should go to infinity as you approach the big bang, thus meaning the big bang was infinitely far in the past. Note this isn't really time dilation (though maybe it could be reframed as such) which the comparison of time like coordinates between different spacetime coordinate systems that are set up to represent different observers' points of view.
In fact I was thinking about this the other day and I think this argument doesn't hold water. It is more sensible to talk about the number collisions in a given comoving volume and use the conformal time as the coordinates, because the number of collisions in a unit comoving volume per unit conformal time should be constant throughout the life of the Universe. As @Stan Liou alludes to though, in the models of the Universe best thought to represent our reality the big bang happens at some finite past value of conformal time.
