First of all, it won't be the end of the world if she doesn't make it and going up 3 levels is quite a jump. Personally I would set 6 as a realistic target as far as you and she are concerned and not worry too much about the school target.
The 4a she got will be based on a Key Stage 2, 3-5 level paper, the 4b is probably based on a Key Stage 3, 4-6 paper. If you have a look at an example of each (you can download from internet), you'll see the content is different. If she is not a confident mathematician, she may take a while to adjust to new things such as algebra, more advanced calculator work and trickier shape work. Thus she may have performed at 4b level on that paper but still be able to produce a 4a or even 5c on the primary paper with which she is more familiar.
My own son was a 4B at primary school, went to a fairly mediocre comprehensive which set him a target of 5 for Year 9 which I thought low (although this was not personal to him as the whole of set 3 was given a target of 5), he scored a 6 with some help from me and eventually went on to score a B at GCSE with a significant amount of help from me. If your school is setting 7, then they clearly are ambitious for her and, trust me, that is better than having low expectations.
Just keep trying to bolster her confidence and help her if you can. Confidence is very important with maths because it is such a logical subject--if you get in a 'tizz', then logic goes out the window.