Motivate students to work on exercises if solutions are provided
The way for a &quot;not so good&quot; student to become better is to work harder and solve more exercises. Reading a solution is not at all like finding a solution. Your test becomes one of memorization rather than skill if you test on exercises for which the students have already seen correct solutions.
I would suggest that if you provide solutions, rather than individual help on finding solutions and seeing where the students went wrong, that you post them after the test.
And, the key, post those answers with additional exercises for those who did poorly on the test to get some additional practice. If you want to incentivize them to solve those exercises, provide some bonus to the test scores for submitting correct solutions to the follow-up exercises.
But you have an additional problem in that your &quot;practice sessions&quot; are probably ineffective if they depend on students knowing that their supposed solutions are wrong and that they need help. Students have misconceptions often enough and need to be pushed past them. If they &quot;solve&quot; exercises but no one checks the solutions, then they are unlikely to see where they made a mistake. The exam only tells them that the are wrong, but may not indicate why and it probably won't help them dispel those misconceptions for the next try.
The combination of a bit of stick and a bit of carrot might help. You have the stick (the exam) but you are missing the carrot. Give points not just for success but for hard work, even if it is re-work.
I often learned lots of math by doing lots of exercises beyond what was required. It was only through that extra work that I was able to achieve insight. Luckily I was self motivated in those instances, but not everyone is. In another course the professor provided lots of external &quot;motivation&quot; through &quot;pop quizzes&quot; which we hated but knowing they were coming, forced us to be ready every day. Remember &quot;flash cards&quot; from early elementary school? Lots of repetition. Lots of feedback.

An option that I've never used, but might consider in your situation is to require that students who did below a certain mark on one weekly quiz, submit their work for review in the following week (or two). This might reduce the number of papers you need to give feedback on and also make it feel like a reward for the better students who don't need to submit. Or even, make submission optional to everyone. And comment on the papers you do get.
With TAs, have them comment on exercise papers and pass them to you for review.