Technically the system is pay-as-you-go in the US; employers are required to withhold estimated tax from paychecks and pay the IRS in the employee's name, or quarterly payments are made if self-employed. The broken part is that you will receive penalties if you withhold too little, or you end up loaning cash to the IRS until you file for a refund; and there are far too many variables that change the actual tax you owe for the year, so there's no way to get the number exact except in limited circumstances.

A "cheap" fix would be to eliminate deductions, and automate marginal withholding. An expensive fix would be to service deductions on a rolling basis, applying them to future income withholding.