How can I improve undergraduate students' writing skills?
Several thoughts.

Give writing assignments (goes without saying). Probably more short
ones rather than fewer long ones.
Make sure students know that form matters as well as content. To
encourage students who start from a disadvantage, count improvement
as well as mastery.
Collect and critique first drafts before grading the final work.
Allow students to resubmit essays.
More problematical: have students critique each others' work before
submission. Those critiques could even be private, so that students
wouldn't be calling each other out before the teacher, only noting
when some argument was unclear to them.

Of course this will be time consuming for you and for them. To do it properly you may have to revise the course objectives to include work done to improve writing - at the expense of &quot;more content&quot;.