I did undergrad out in the UC system. There, there was only one quarter-long course before calculus. So in 9 weeks (10 counting finals), we covered all of precalc - college algebra & trig. If a student needed any earlier math, they could take it at a cc. I think that's appropriate.
It will vary from school to school whether precalc is considered "college" level or not. I vote not, however many schools will give college credit for it.
Look at course descriptions to see what content is in each class.
Where I'm teaching now, Introductory Algebra is basically high school Algebra 1. We review basic arithmetic (sigh), cover solving linear equations & inequalities, graphing linear equations and inequalities in 2 variables (along with slope & equations of lines), exponential arithmetic, polynomial arithmetic, scientific notation, factoring & solving quadratic equations by factoring.
Intermediate Algebra is basically Algebra 2. We cover arithmetic with rational expressions, solving rational equations, arithmetic with radicals, solving radical equations, solving systems of equations (2 variables, sometimes 3 depending on what text we use), review of graphing, generally including graphing absolute value functions, generally some talk about functions, occasionally discussing function arithmetic and inverses (again, dependent on text and timing), and graphing parabolas and solving quadratics by using the quadratic formula.
College Algebra includes rational roots theorem, graphing by translations, exponential and logarithmic functions, and depending on timing, conic sections.
Trigonometry includes trig.
And then you finally get to first year college level work.
I'm teaching developmental math... prealgebra.
I would be amazed if my 10 yo weren't able to pass my course.
But some students really need to start at a lower level. I'm just dreadfully frustrated by the students who are straight out of high school - who have a hs diploma - and then test into a course that is lower than they can get high school credit for.