As someone who lives across the pond, it never ceases to amaze me that this question needs to be asked at all. Is tap water safe to drink? I wouldn't take "no" as an answer unless I'm in some remote third-world country that has known issues with the availability of drinking water. And yet...
Not saying tap water here tastes great everywhere (far from that), but you can be pretty sure it won't cause you any physical harm.
> Drinking water is chlorinated and has no ill effect on the local population. However, some strains of e. coli (naturally occurring bacteria found in your gastro-intestinal system) may be present in very small concentrations in the local water supply. Some local strains are different than those that you may be used to, and may cause diarrhea in travellers since immunity is not developed as a result of short-term exposure. Using bottled water for the first few weeks will help you adjust and decrease the chance of traveller's diarrhea.
https://www.iamat.org/country/italy/risk/food-water-safety-o...
Poking around, the same warning is in place for Spain and Portugal.
I always get bottled water when travelling, although I tend to get the cheap local or store brand water. A few euros is really cheap anti-diarrhea insurance, and it tastes a lot better than chlorinated water, too.
That being said, for travelers it sure might take a while to adjust to local conditions. I never had such problems when traveling to Spain, Austria, France, Hungary and Slovenia, but I do understand that each body copes differently with things it's not used to.