It is in fact illegal for public schools in the United States to compel students to participate in religious worship or practices. It has also been upheld in the US Supreme Court that teacher-led prayers cannot be held in public schools because it is either exclusionary to non-participating students or marginalizing to those that do participate (depending on how it is presented). In the US, there are public and private schools, and some private schools are religious schools. This definitely happens in religious schools and is legal there, but it would become an ethics dispute if a public school attempted it. I think the closest I got to any forced religious exposure in my time at school was visiting the Washington National Cathedral, but that is also a place of architectural & historic significance in the US because some important individuals are buried there, like Woodrow Wilson (a former president) and Helen Keller. It was part of a voluntary trip to the capital, a relatively short part of the visit, and a service at the cathedral was not part of it.
Grew up in the states. Went to public school all my life. In kindergarten our teachers took us on a field trip to a catholic church without indicating they would be taking kids to a church on the permission slip and sat us all down in pews and had someone come out and say a few words which I don't remember. During other periods of schooling, we were taken on field trips to missions, which are also churches, and has actual historical value and was conducted in a completely different context as opposed to the catholic church during kindergarten. Personally, I didn't mind it since I didn't see any religious indoctrination and thought it was a just a cool experience. You best believe we were never taken to any other type of place of worship, e.g. jehovah, latter day, protestant, jewish, hindu, muslim, etc.
Perhaps to give a little insight into the situation in Turkish mosques, although I can't see the picture and I'm not exactly sure about the rest of the context... it is forbidden for any tourists to enter any mosque without first being properly covered and robes or scarves are provided if they are wearing shorts, tanktops, etc. It probably has a lot less to do with 'indoctrinating' students than it does with showing 'respect' to the tradition of how people have been entering these places for 1000s of years and that is with covered knees, elbows, and womens' hair. Under no circumstances would any of the viewers be taught any religious dogma, prayers, or whatever, as that would go completely against the nature of islam, and in fact, out of respect for worshippers, I'd be pretty certain that they wouldn't even be allowed to view the prayers. Although it wouldn't surprise me if things are slightly different nowadays. Personally, I would like to see the same type of activity where turkish kids would be taken on a field trip to a church but maybe that step will hopefully come in the near future. I'm not sure about american kids today but I really can't imagine that any of them are being taken to mosques and at any rate religious homegenity in turkey far surpasses that of the united states so it's a completely different sort of deal.
Edit: just saw the picture. Holy cow the optics of that are pretty terrible. I can definitely see the kids being preached at there and it seems it goes way beyond polite covering up. Their T-shirts are saying something to the effect of “the apple of my eye is prayer”.
@Kentucky James VII canli bomba means “bombshell live breaking news”.