speed of chess improving

A  friend told me that 2  or 3 month are  enough to get  a rating of 1600. It's  already 1  month that I  played and I'm  still a 1300 so I'm worried that I'm  progressing too slowly. My goal is to get the  title of Fide master in less than  three  years. Is  that possible? Or am I unrealistic?  How many years of training that it  required to get  a rating of 2000?

Your friend was wrong.   A 1600 rating is better than 90+ % of the players here. 

Do you think that everyone can beat 90+ % of the players here with only two months of work?

That is ridiculous.

A 1300 is a very good rating after only two weeks.  It seems likely to me that you already knew how to play before you joined.

@1

"A  friend told me that 2  or 3 month are  enough to get  a rating of 1600." ++ Correct.

"It's  already 1  month that I  played and I'm  still a 1300 so I'm worried that I'm  progressing too slowly." ++ Do you blunder check before each move you play?

"My goal is to get the  title of Fide master in less than  three  years." ++ That is ambitious.

"Is  that possible?" ++ Yes, but not probable.

"Or am I unrealistic?" ++ Probably.

"How many years of training that it  required to get  a rating of 2000?" ++ 1 year

Here is an example of rating progress from 1600 to 2000 in 1 year
https://ratings.fide.com/profile/240990/chart 
Progress to 2300 FIDE master level took another 2.5 years.

yes, I  learned chess when I was kid. Back then I remember not a single one of my friend could beat me but after that i stopped  playing during many  years. Recently  I just witness how  strong  people  on  internet  are and it  awakened again my passion for the game.

Yeah i've been playing for two months and am only 1100 (I started from 600)

No.

Like @justbefair said, that's ridiculous.

And it took Kasparov and Karpov ~2 years to get to 2000, so as usual your other rating information is is also ridiculous.

SOME people are able to quickly improve their rating to somewhere like 1600 in a few months (online of course...) They are scattered few and far between. And then when they stay at a rating for more than four months they think they're stagnating...

In any case FIDE Master in 3 years is setting yourself up for failure. It's best to focus on improving first before you worry about achieving a master title. Unless you're on track to become a contender for world's youngest GM, of course, which you probably aren't.

Yeah ok, 1600 online vs OTB is different... and like most people who visit chess.com, OP isn't a beginner.

Beginners are really freaking terrible... I don't think most people realize... a real beginner's rating would be below zero. If it's the first day you learned chess and you've never played a game like it, then you're going to be rated below zero.

Can the OP get a 1600 rapid rating 3 months from now? I don't know, but at least it's more likely they do that than getting FM in 3 years.

Your friend is lying to you.

First of all, online ratings are not really a good measure because they vary widely by site.  But for the sake of improvement trajectory, let's consider USCF.

Here is the rating graph of Hikaru Nakamura (who was considered a child prodigy):. https://www.uschess.org/datapage/ratings_graph.php?memid=12641216

As you can see, it took him about 3.5 years to go from 600 to 2000.  Note, that he played unrated for about 2 years prior to that initial rating of 600, so effectively it took him about 5 years to get to 2000.  That is pretty fast, even for kids.

A local kid is rated in the top 30 in his age group nationally.  He has been playing for almost 10 years (starting at age 6) and just crossed 2200.  It took him about 4 years to reach 1600, once he started playing USCF rated games.

Someone that tells you it is possible to reach 1600 in 3 months of playing is feeding you nonsense.  1600 is an above average club player.  They have a decent understanding of opening principles and the basic tactical patterns.  3 months is not enough time to accomplish that.  It would be like telling someone who has never exercised a day in their life that they should be able to bench press their own bodyweight within 3 months.  Highly unlikely for virtually everyone.

1300 is an EXCELLENT rating for a relatively new player.