B.K.
You are describing how my older daughter used to be. She struggled through math (and a couple other subjects). She would do fine with our help at home or with the tutor, and just blow the tests.
It took us until her junior year (poor thing) to finally get the right help for her (through public school not private) and we found out through extensive testing that she has ADD-inattentive type. So she is very, very distracted by sounds, pressure (such as a timed test) and other activity in her immediate area. Tests were a nightmare because of this. She ended up taking tests alone in her counselor's office at school for her last two years of school. It made a HUGE difference.
Also I want to mention that she was in private school for many years, and then we switched to public. In our area, the private schools are much harder academically than public. Perhaps your son is having a difficult time adjusting to that, and to other kids and a new environment, etc. See if you can get him some help through the school -- testing for learning disabilities, etc.