Let’s break this down a bit.

As the matter of fact, I think not only Dostoyevsky, but most of the other big classics of pre-20th century wouldn’t be recognized.

Of course they wouldn’t be recognized. Recognition of greatness is something that takes time. A ‘classic’ isn’t something that just pops off the printer and hits high sales, and even a lot of what we normally would consider part of the ‘Great Works of Literature.’ had more than their fair share of criticism on release. Next to no novel ever published was considered a pinnacle of literature shortly after being published, or least not with any seriousness that wasn’t a sale pitch part of a publisher’s ad campaign.

Time is also a pretty good filter. ‘People wrote better back then!’ is bullshit if you consider the hundreds of thousands of novels that have been written, read, and absolutely forgotten, and often (though not always) with good reason.

popular genres now are; fantasy, young adult, post-apocalypses, thriller, biography and others, have nothing to do with most of the classics written.

First, most of that list are considered subgenres. Second, most novels fall under several different genres. Don Quixote for example is considered both satire and fantasy, pretty easy to categorize that one. Romance novels have dominated sales since we started tracking sales, and still do today. ‘But everyone is reading YA today!’ No they’re not. They’re still reading mostly romance schlock and by the truckload. They’re not talking about them, mostly because there is little to say. Most sales charting puts classics within their own category, rather than include classical literature in another genre, so it’s not that they don’t fit into the general broad genres, it’s that we don’t usually do it with classics. For the why, you’ll need to talk to publishing companies to get a definitive answer.

Secondly, it feels (and maybe I’m wrong) that people now, prefer more fast-paced and filled with action books. I very often hear and read complains how certain books like “Crime and Punishment” or “Anna Karenina” are plain boring, and there is nothing going in them. As I see it, their is a tendency to value real physical dilemmas more then psychological ones.

You are wrong, but also right. People today prefer page turners, if we go by just general popularity of sales figures, but surprise, people of yesteryear did too though. No one has ever accused Dostoevsky, Tolstoy, Bronte, or Fitzgerald of writing page turners. They wrote terrific tales woven with masterful skill, but exciting? Not really. But the exciting page turners, or even just genre trope novels, usually don’t stand the test of time. Plenty of books were written, read, and forgotten and during their short life they probably outside something that today we lump in some top 100 list. It’s all too easy to look at classical literature and think that everything written at that time period was on par. It wasn’t. Bawdy tales, bad satire, and the literary equivalent of a fart joke were often the contemporaries.

And lastly, taking the risk to sound like an old man, it seems that less teenagers and adults become interested in reading classics. If so many people are already refuse to read some big classic, there wouldn’t be a chance for it if it was published today.

Today, not that many people are reading Dostoevsky. Twenty years ago, not that many people were reading Dostoevsky. Fifty years ago, not that many people were reading Dostoevsky. No one was really reading Shakespear for like 150 years. In other words, there might very well be a terrific author that everyone is going to be talking about in 2255 who published this year, and both you and I might not even know about it at all. Even worse, we both might even read it and think ‘Well this book is pretty meh.’ and forget about it entirely. None of this is because of any great change in society, or because of ‘Kids today/people today/society today followed by whatever problematic fault is perceived to only exist now but actually has been around since we were huddled in caves.’