Employee put as an independent contractor in taxes

In terms of income taxes, you don't owe any more in total as an independent contractor than as an employee: only the timing is different. When you are an employee, the employer withholds part of your salary for taxes and sends it directly to the government; the money sent to the government comes out of your wages, so your paychecks were smaller. As an independent contractor, you get all your wages upfront, but then pay your share of taxes later. To oversimplify: say that based on your income, you are paying 20% of your income in income taxes (tax rates are based on income). If you earned $10,000 as an employee, the employer sent $2,000 from your checks to the IRS as income taxes and you received checks for $8,000; if you were an independent contractor, you received all $10,000, but then you later have to send $2,000 to the IRS. The income tax is the same: only the timing of when you pay it is different. The problem you have is that you did not budget to pay those taxes: if you had, you would have put around $2k aside and not spent it, to use for taxes. But since the amount of income tax is the same and is owed by you, not the employer, they are not liable for this; it's your income tax payment, whether paid on the front end or back.
There is a difference in terms of the much smaller social security, medicare, etc. payments--for an employee, the employer pays part of those amounts for you, whereas for an independent contractor, they do not. Potentially, if you believe as you do (and based on what you write, you may well be right) that you should have been characterized and paid as an employee, you could bring a complaint to the state or federal Dept. of Labor. If the Dept. agrees, it can order that  the employer make those social security and medicare payments--the employer portion of them--for you. But that will not affect the largest part of taxes (income tax) which you owe.