How big is your tank? What temperature do you keep it at?
Try increasing your maintenance up to a small water change every week end (10-15%), wash out the filter sponges in the old tank water whenever you do those.
If your palm looks like the photo below, it is not a true aquatic plant and will die with time, but not to worry, it should survive for a while yet and I'm sure you will be able to find something to replace it with

I would advise that you try something like Elodea...
"Aquatic" fernAll in all, algae is (IMO) natural and you will find some in almost every home aquaria and not all are edible by "algae eaters".
p.s. plecos appreciate a spot of courgette or cucumber every now and then weighed down with a stone & rubber band / something heavy