Give it another few weeks. Try a scratch test on several twigs/small branches of the questionable ones. If the cambium is still green, then it's more hopeful. Make sure they have water, but don't drown them - and don't feed them either. If the cambium is brown, go further down the trunks/branches until you hit green. You can trim off the truly dead stuff, but be careful not to damage any possibly live stuff - it may be better to wait and be sure it's dead before you cut anything. Cut off the dead trunks at or just below ground level. The tree/clump should have a good root system by now, so if the trunks all are alive, they should go well, once they get going. If the two are dead, and you really want a multiple-trunked specimen, then you can select 2 or more strong suckers from very low on the base of the live one, and let them grow over the summer. Next year, you can select the strongest one(s), and go from there. It will take longer, but you will have a multi-trunked tree.

I did cut of the trunk down to the highest water sucker growing out of it. First did a scratch test and the trunk showed no green, but when I cut it off I could see the slightest bit of green beneath the bark. Maybe I should have waited until that part of the trunk turned brittle, but now at least the tree can at least put its effort toward making food and feeding its roots with those lower water sucker branches that are growing out of the trunk. Then maybe next year the roots will be established well enough to send up some shoots from the roots and I could then cut the old trunk down and allow the tree to focus its efforts on developing new multiple main leader trunks that should have better balance between them in trunk diameter.

You did not say how long your Birch trees have been growing on the property. If they are well established, you most likely are not considering how wide spreading birch root systems are. If you are putting a hose on the tree, that would not provide the needed water as quickly as the trees need. It would rather be creating a situation where the area around the hose, or where ever gravity draws the water, is watered enough to become deep watered, but the trees wide spreading roots could still remain dry. Additionally very few of the roots will benefit from the deeper soil moisture. Birch roots grow to the depth of the first foot or two of the soil, and they can spread across areas as wide as two, three, or more times farther than the birch's canopy spreads. It would be better to use a sprinkler system that is designed to broadcast the water; not only around the base of the trees, but also around the extended surface area where the trees roots may have spread. Another thing you might need to adjust is how soon you take note that the trees need watering. Do you watch and water at the very first signs of stress the trees show; or do you only jump into action, trying to respond when you see yellow leaves on the tree with some being shed? If you can respond to your trees need for help, before their damage results in the trees going into a kind of shock, then they should bounce back much more quickly.

I have an approximately 50 year old cutleaf weeping birch and it has dropped small twigs/branches for as long as I can remember - not nearly as many as a poplar or willow but always something on the lawn especially after a windy day. It`s not that much of a problem for me but then it`s only 1 tree as opposed to your 4.