but I think it’s a very American idea that media and public discussion about a case jephorises the case by tainting the jury somehow
No, I think it's a very British idea that EVERYONE is entitled to a fair trial no matter what they are accused of. That's why, once someone is charged, all speculation/rumour/gossip by the media should (and in 99% of cases does) stop and comment are closed on newspaper articles as soon as charges are brought and whilst trials are ongoing. You can't really do much about social media unfortunately.
The USA, on the other hand, clearly don't give a
duck about fair trials given the amount of publicity and information they allow before a case gets anywhere near a court room. Take, for example, the latest obsession - Lori & Chad Daybell. I think we're at three Dateline episodes and at least one 48 Hours so far (to my knowledge - there are probably other similar shows covering it) and that's not even gone to trial yet. But then, beforehand and once it starts, you get 24/7 rolling coverage on Court TV, gobshites like Nancy Grace arguing it out every night live and a jury that's not sequestered no doubt so can watch, read and talk to whoever they like.
And as much as I would love to watch the trials I follow here, I'm very glad our justice system is not a media circus. There was a programme on BBC2 about the murder of Margaret Fleming in Scotland and filming was allowed (I believe they do allow filming of sentencing remarks and appeals in high profile cases there - I think this was maybe a trial run (no pun intended) for filming an actual trial) and the show featured important parts of the trial including witness testimony but it was shown way after the verdict was delivered. I would be in favour of something like that.