While there are more expert folks here, I'll take a go at your question...

If the tree is only 5' tall after 3 years in the ground at your place, it is not growing that well. You also mention that you allowed turf grass to grow up to the trunk. which may be a factor too.

We probably need some more info too, such as what type of rootstock the tree was grafted to (full size, dwarf, sd, etc), whether you've checked for borer damage (which can stunt or kill young trees), as well as any signs of abnormal growth (discolored leaves, dying branches, etc).

However a few things to think about: Are you using weedkillers on the lawn around the tree? Is this a dwarf tree, which does not compete well with grass? Signs of borer or rodent damage underground? How much exposure to sunlight does the tree get? Water?

I would try and figure out what the cause(s) are of slow growth before I began taking branches off. It may make the situation worse.

Whenever a tree is transplanted, it loses about 90% of its original root system. It is for this reason that most experts agree that whenever you plant a tree, you should prune it back severely at planting to about 2-3 feet high so that it will remain in balance between growing new roots and the top of the tree at the same time. Unfortunately, it sounds like you did not prune when the tree was planted several years ago. So then it has taken the last several years to grow all the roots, and hasn't done much of anything in the top of the tree. If you want to see real growth, I'm sorry but I wonder if it might do the tree a lot of good if next spring if you just chop it down at about 3 feet off the ground. Not to worry -- it has indeed spent the past several years growing a lot of roots, so I bet the tree will bounce right back and start acting more normal again. Alternatively, you could just leave the tree alone and keep on hoping and praying that next season will be the one where it's done growing roots and starts acting normal. Maybe it will be. But the best guarantee might be to lop off the top next spring. Seriously. Don't do it right now in summer/fall as it is most likely too late in the season for it to have time to recover.

Best of luck to you, whichever route you decide to take.