Do we bear sin if there are websites on our servers that violate Sharia, which we only discover to be in violation after receiving a complaint, or if we design websites where their owners later place things that violate Sharia?

If websites are handed over to their owners, there is no sin upon the one who hands them over if he is not certain or does not predominantly believe that their owners will use them for unlawful purposes. As for hosted websites, conditions must be set to prevent any act contrary to Islamic law on the servers. If any violation is noticed, their owners must be enjoined from evil and commanded to do good. As for what has not been seen, there is no sin upon the host as long as he did not know about it or approve of it.