Is it normal for my contacts to make me look like I have a lazy eye?

No. If you have an underlying tendancy for an eye to turn in or out when tired, an improper prescription contact lens may bring this out. Otherwise, it would be unlikely for this to occur from contact wear itself.
Lazy eye. Can;t answer becuase I need to know what you mean by appearing to have a lazy eye. If you mean crossed eyes then it would be very unusual for contacts to cause this. However if you have a vastly wrong prescription or if you already have crossed eyes that you don't notice when wearing glasses then wearing contacts may make that evident.