Are there math graduate schools without stressful tests? I am one year from graduating and am starting to look for Graduate programs. I would like to know if there exist graduate programs where the focus is on research from the start (apparently this is the case in the UK).

I ask this because I have noticed I learn better when motivated by research than in regular courses. Additionally, because stressful long written exams frighten me and I would prefer to avoid them if it were a possibility. It would be nice at least to go somewhere where they are more flexible about exams, and place less value on them.

I ask this question, not seeking for a list of schools I might consider applying to, but to get a better idea of what graduate programs loook like. I would like to know if it’s worth it for me to focus on this, or if I should just look at the academics because focusing on this would be too restrictive. ( I cannot live by avoiding stress at all costs as mentioned in the comments).
No one gets a PhD in mathematics (or any other subject) without experiencing stress, period. But there are different kinds of stress. Exam stress and research stress are different, and some people respond better to one or the other. In particular, the time constraints in research are obviously much much more flexible.

As Nate Eldredge notes in the comments, pretty much all US programs have some form of qualifying exam. UChicago is a notable exception, but their first-year graduate courses are more difficult, to make up for this. (We might formulate this principle as "conservation of stress.") But the important thing is that even though students freak out over qualifying exams, these exams are merely a hoop to jump through, and are not the main focus of any PhD program. Don't get me wrong, they are a useful hoop, but once they're over...they're over. You will stop stressing over exams, and start stressing over your thesis. Typically, qualifying exams happen around year 2 of a 5+ year program--the remaining years are all about research.

To put it another way, avoiding a good PhD program because of exams would be like avoiding a good vacation because of a fear of flying: perhaps necessary if you have a true, diagnosable phobia, but otherwise, you should grit your teeth and get through it.