From a practical perspective, how do you expect your employer to pay you if they do not know the hours that you worked? It definitely sounds like you work on an hourly basis instead of a salary. Why are you balking at turning in your time cards? As someone who used to prosecute employee embezzlement and fraud, the reluctance to turn in time sheets or time cards were huge red flags to us that something wrong was going on. You need to step out of your shoes and into their shoes to see how this would come off to them and simply provide them with the time cards. That is simply not too much to be asking for.
The information provided in this answer does not create an attorney-client relationship and is not considered to be legal advice.
As noted by my colleagues, employers may require that you turn in a time card in order to process payment of wages. To th extent that you might be a salaried and properly classified individual, even then an employer may withhold payment, if their policies are clear about the need for the time card (e.g. in order to invoice clients).
All communications are provided for general informational purposes only, may not be applicable to your specific set of facts, and do not constitute legal advice or a legal opinion, or form an attorney-client relationship. You should not act or rely on any information provided through this website or the Internet in general without first obtaining legal advice from an attorney specifically hired to represent you on a given matter. For more information, go to www.theARgroup.com.