The hiring process for a postdoc position in the US I need a piece of advice.
I found an interesting postdoc position one month ago. I applied all required documents (CV and cover letter) and 4-5 days later I was interviewed by the group (including head). A few days later I was asked to prepare the research highlights and I did that. One week later they asked me to apply for the position via the university website if I am interested in the position, and also requested recommendation letters from possible referees. All letters were provided as I know, and they are positive. Moreover, I know I well match to that position. Of course, I do not exclude other possible existing candidates for the position and I can be not the first among them, of course. So, one month passed from the interview.
I haven't experience in the application process for a postdoc position in the US and this is for the first time for me. Thus, the evaluation and hiring process for a postdoc position is a bit unclear for me. I am not sure that a direct request to the lab head for clarifying the situation is a good idea because in this case, I can look like a hysteric or in need person, and I don't want that due to obvious reasons. I do not exclude that they must prepare for the position a justification or something like and this also needs time.
In accordance with the above, I need a piece of advice. What should I do? Is the situation normal or I'm screwed? What should I choose as a strategy (wait or write)?
The academic decision clock can run very slowly. If you don't have other options then I'd suggest some patience. There may be a number of interviews they need to have with others and there may be issues about scheduling decision meetings. Lots of things can slow it down.
If you have a specific need to know about the decision then you can be a bit more aggressive in asking, but otherwise, a request for an update after a couple of weeks is probably acceptable. The most information you are likely to get, however, is that you are or are not, still in the running. You can ask whether they can give you an &quot;estimated decision date&quot; and they might answer, but  don't expect it to be especially accurate.
But don't pass up other opportunities as you wait.
I doubt that very many places will tell you that you aren't the first choice and if others are ahead of you they will wait for a decision from one of those before telling others they won't get the position. But don't read too much into a delay, and don't panic. There are too many possibilities.