I don't think that I'm particularly well qualified to comment on the average European's view of Israel/Palestine, but I can comment on how things are here in the UK. If we look at the main-stream media outlets such as the BBC the pro-Israel bias is reflected in the manner in which news is reported but is not overt except in the case of the right-wing newspapers. I really can't imagine a newscaster talking about Israel as "a partner against terrorism" or saying that our country has "the same values as Israel." Respectability is a key imperative, and whilst media outlets won't want to be seen to be sympathetic towards Hamas or Hezbollah neither will they want to be seen to be sympathetic towards Israeli 'excesses.'
So even within the media there is a huge difference, but when you move on to public opinion I think a really significant difference starts to emerge. The BBC faced a huge back-lash over the manner in which it reported the latest round of 'fighting' in Gaza, with many members of the public, with no political agenda, feeling genuinely outraged about Palestinian fatalities being reported in a completely different way from Israeli fatalities. Although Britain is home to the most famous royal family in the world we also have a long history of class-struggle, and although it is no doubt something of a generalisation, it is my impression that many working class people identify with the Palestinians as an oppressed population group, and as such genuinely desire their release from Israeli oppression. There is a sense in which working-class people in this country feel personally threatened by the privileging of Israelis over Palestinians. Of course most people are fairly apolitical and aren't paying much attention; but in the case of somewhere like Scotland, which has its own history of resisting English colonialism, I would say that this would pretty much be the consensus opinion - in so far as people held an opinion at all. [Google 'Scottish Palestine Solidarity Campaign' for more info on that].
I'm sure that there is a difference between the different generations, and in general I suspect that the younger generation are less sympathetic towards Israel. Of course we have a different history with Israel to the US. I recently met an older guy who was shot at and nearly killed by Jewish terrorists prior to the British withdrawal! It doesn't help that they are still teaching WW2 and the Holocaust to kids in school as if these were the defining instances of injustice in human history, rather than cautionary tales of how careful we need to be to avoid stigmatising or scapegoating any population group under any circumstances, from which we can learn universal moral lessons. However, in general I think that the mood towards Israel is one of apprehension. People who want to offer unequivocal support for Israel are definitely the exception rather than the rule, and likely to be viewed with suspicion by most people who are following developments in the region.