I find that for birchbark, soaking the pieces in warm water to soften, pat dry on a towel, and use a good glue like Tightbond-II . Glue the whole stack up and press while it drys.
Here is a tip/trick:
Punch the center of each piece with a leather hole punch. If the bark is stiff and dry, do it after softening for a while (this prevents splitting).
Take a smooth piece of brass or steel rod that easily slips through the holes, wax it with several coats of car wax, and use it as a mandrel to assemble the bark squares on.
Take a cheep woodworker's clamp and drill holes through the jaws to allow the rod to pass through.
https://www.harborfreight.com/hand-tools/clamps/12-in-ratcheting-bar-clampspreader-62123.html
Soak the bark in a large bowl of warm water until just pliable.
Put some glue in a shallow bowl and dip each piece in the bowl, wipe the excess off with your fingers, slide on the rod ... repeat until all are on the rod, then put in the clamp and tighten down just snug enough to be firm. Let it sit for a few minutes and snug it down a tad more.
Let it dry and cure for a day, then remove and tap the rod out by setting the stack on a board with a hole drilled through it and hitting the upper end of the rod with a mallet. It should tap right out, but if needed, turn around and repeat from the other end.