Can you homeschool through a school district or like online in europe?

As far as I know there is no way to homeschool through a school district in Europe. There are a number of online schools based in the UK and one online school that I know of in Spain. Have you considered just enrolling in one of the many American online schools and correspondence schools that enrol students from overseas? School is a very different animal in Europe (it's hard to see how you'd fit into the german school set-up for just six months and virtually impossible to see how you'd fit into the structure of the french system, for instance) and if you're only going to be in Europe for six months it might be easier, both in the short term and in terms of re-entering the American school system, to stick with what you know, i.e. the American education system.

I don't know about outside the UK, but there are a number in the UK you can use. My son went to Briteschool, and it was fine. He's just done his IGCSEs. There are two others that I know off, first college and inter high. Inter high looks quite formal to me, and first college is very casual. The thing we liked about Briteschool was that all the teachers were qualified teachers, something you didnt get with first college, when we had our son there.