I've found it pretty common for private school in our area claim to be a year advanced of public schools. However, when we were researching more challenging options for our eldest, we found it wasn't the case unless you were comparing them to a handful of lower performing inner city schools (of which most other public schools were a year advanced of them too.) I'm not saying there aren't private schools where this holds true, just that we never came across a private school that was able/willing to accomodate advanced children as well as our local public schools.
As for Abeka, it is heralded as being challenging but I must say, none of our homeschooling friends with gifted children use it (and we know many.) I believe Abeka uses a "spiraling" technique. This is nice in that it gives youngers kids who may be ready for advanced math a "taste" of what is to come but not so great in the large and steady stream of review work that comes with it. An accelerated child can go bonkers having to continually go back and review material they have already mastered!
I will say that 4th grade in our district is a big year for research projects, reports, written essays, ect. Neither of my kids did much in the way of worksheets past 3rd grade.
As to what to do, well, if your kids are happy and healthy, it may not be an issue. If they are becoming frustrated with the lower level work and the school is unwilling to work with you, it may be time to try someplace else.
Married mom of two, DD 17 and DS 14.