aggressive co-worker and bullying at the lab I've recently started doing postdoc in an European country. 

Since the first day, I am having problems with working in the lab environment. The research group is too individualistic (there is no shared protocols, project and any kind of support) and some people are behaving in quite toxic ways.  One of the lab members doesn't even look or talk to us (me and another student) since we arrived here. He ignores and turns away whenever he sees us, as if we are not worth it. 

The main problem is the postdoc though. He is too assertive, aggressive and sort of being a bully in lab meetings, trying to get involved in every project and confront you without significant reasons. He even interrupts when he thinks it is boring and not delivered very well or wants you to prove (show RAW data) when he doesn't think it is enough. The PI does nothing and she is so dependent on him, lets him to be a co-PI or the kind of bad boss. 

So, basically, I don't know what to do. I know that I can't stay it more and be productive here, but I am in a small country, where all the academic world knows each other. 

Do you have any suggestions about finding another position or doing applications at the same university? If you could give some ideas, I'd be very grateful. I totally feel paralyzed.
It is unlikely you can fight this culture, and academic settings have practically no good ways to address bullying of this nature. If you find what seems to be one, look for evidence that it has worked for past students. Recently.

Alternately, make the bully your friend. This is not advisable.  Alternately, work with people the bully cannot impact, which is a version of leaving. This may be unavailable, depending on your situation. 

Calling out this bully may help, but if it does not it will make you a target.  Again, it is highly unlikely your institution has any kind of meaningful recourse available.

Your best strategy? Leave. Keep your head down, and start planning your exit now. After you have an exit, cite the bully as the reason to both your PI and HR; it may help others.