This is a completely normal phenomenon and is not something you should view as a personal deficiency of some sort. Generally speaking there is an axis at one end of which there is the ability to deeply concentrate and analyze complex matters and at another end of this axis there is the ability to eloquently talk. People are all different and find themselves at different points along this axis, with a varying ratio of those two skills.
Some people can't stop talking but they can't really do any professional work either. Others aren't good at talking but possess the ability to build castles and empires in their minds. While I'm sure there are bright individuals gifted in both of those skills, it's been my observation that those two are mutually exclusive. Not in a black-white fashion but in a way that an advance in one skill implies the other has somewhat suffered as a result. More like two positive fractional numbers that make 1 when combined but can't both be 1.
Note that I'm not talking about the "offline" ability to express one's thoughts like in writing, let's call it this way. People who are clear thinkers are often great communicators or at least have this ability in themselves they can develop if they set their mind to it. I'm talking about this ability of thinking and talking at the same time that you mention. It's rarely found and cannot be expected from everyone.
This cleared now back to your immediate problem. First things first you should not panic or feel frustrated, no more than about the rainy weather (if you prefer sunny, otherwise it's the other way around). What you should do is communicate to your interviewers what kind of person you are, that you can easily discuss working matters with your colleagues but then when there is some thinking to be done it can only happen in silence. My guess is that you can't work in an open plan office either so these are the things you need to communicate to your recruiters/interviewers. One bit of advice: do not word your communication as if you were apologizing for that would-be limitation of yours, but turn it around so that the other side feels they are somewhat deficient in their processes and their way of working and can't quite live up to your standards. If they want you on board, they might adapt the interview style or skip some parts of it entirely.
I see you describe this style of interview as "the norm". Indeed it has become ubiquitous but is not something I would describe as the norm. I wish to point out that this only happens because developers accept to be treated this way. If you want these practices to go away you have to push back. I for one haven't done sample assignments, technical interviews or brain-teasing problems for many years and that is something I communicate very early in the negotiations if somebody wants to "interview" me. Getting to know each other and discussing potential collaboration is good, being examined is not. For that you have your academic degree (and hopefully some great projects to show). This has always been a good filter to discard places and people I do not want to work with. An interview should be used wisely to discuss things which are really important, to get a sense of each other's personality and to find out if there can be a match and a productive collaboration can take place. Other than that it would just result in a few wasted hours in the lives of everyone involved.
Long story short, it's all fine, cheer up, adjust your process slightly and move on. Better opportunities will come.