The reason college is expensive in the US for some is because the government pays for large portions of the population to go there. 85-90% of college students have some form of assistance which means they pay very little to nothing out of pocket. Only if you make significant income (>$250k), do your children not qualify for government assistance.
And for those in Europe and other places where we get 'free college' - full tuition out-of-pocket including room and board averages ~$50k/year at a government-backed guaranteed ~2% interest rates loan you're not obligated to pay back until 1 year after you finish college. And that's "high", $200k in debt to (if you choose well such as a STEM field), you typically can get a job starting between $50-100k with <18% effective income taxes. A few years later you should be making $100-250k/year. You should be able to pay back the loans with the tax savings alone vs your average European country.
Off course, when the government gives free money, there is no limit to the cost because the government will always pay more, they'll just take it out of your taxes, so the tuition goes up. And with free money, people don't make good choices either, you can get a degree in just about anything for free and as long as you take one class, you don't have to pay back any debts either. So people get degrees in the liberal arts, rack up debts and don't have to be responsible, until they need to get a job and they realize the degree in lesbian dance theory does nothing for them, so now they're demanding the government forgive their debt and give them MORE free stuff and guarantee a job?
I would suggest that public school for K-12 is not very useful either. The better option is to have school choice, vouchers for those that don't have the income to pay for private schools and reduce or eliminate the system of school taxes. In my area we pay upwards of $1B/year into the school system or ~$35k/student and we get dilapidated buildings, class sizes of ~20-30 and the school system has ~$500M in debt. You can pay a private tutor handsomely including supplies for every 4 children with that money.
American college, as European college is still very cheap. Even at those 'exorbitant' tuition rates, the government also pays ~50% of operating costs for any privately owned University through research grants and other funding. So your college should really cost twice as much as it does today, or without government assistance, it would probably cost about as much as it does now, just with less administrative overhead.