I dont understand how hybridization helps me?
Hybridisation is key to the problem.
The system is aromatic if there are 4n+2 electron in a delocalised
pi system. It seems to me that you are getting confused with the "n". The value of "n" can be any integer, it simply means that an aromatic system can have 2 (n=0), 6 (n=1), 10 (n=2), ..etc. etc. electrons in the delocalised pi system. I do'n understand what you are talking about when you say
4 electrons in each C=C bond so 4π bonds
To me, that makes no sense.
Pi symmetry is required, and you need to combine p atomic orbitals to make pi molecular orbitals.
In the case of furan - the oxygen does have two lone pairs, but only one of these is part of the pi system, the other is orthogonal to the pi system - meaning it does not overlap with it (see picture below).
So overall for furan with an sp
2 hybridised oxygen we have 2 pi bonds (2 electrons for each) and 1 lone pair (2 electrons) in the pi system. 2(2)+2 = 6 therefore aromatic.
The situation is similar for 3, and for 4 you can play exactly the same game - I count 4 pi bonds and 1 lone pair
in the pi system (there are 2 lone pairs orthogonal to the pi system).