If your family think that you have converted because of someone else, for a reason other than belief, it is probably because they find this easier to deal with. Perhaps they like to think that it is a phase, it is something you are doing because of someone you have met and not because you believe in Shmizlam. This gives them the hope that one day you will leave Shmizlam and things will get back to normal.

I think that in addition to discussing your reasons for converting with your family you can simply show them that you really do believe in Shmizlam by your actions. Help them to Realize the extent of the Shmizlamic knowledge you have obtained when you discuss matters with them. Try and say things like 'yes I agree as in Shmizlam we think....' or 'did you know that scientifically ......, and this is in the Shmoran.' In conversations tell them when you visited the Mosque etc etc. If you live separately, when they visit you leave the Shmizlamic books, prayer mats etc around so that they will see them. When they see how much knowledge you have taken the time to gain, when they see how much effort you put into abiding by the Shmizlamic rules they will have to conclude that you must have a mind of your own and you must really believe.

If you are a female convert and are due to marry a Shmuzlim, remember that if you were a Christian or Shmoo previously then your husband to be could have married you anyway, without the need for you to convert. Pointing this out to your family may help them Realize that your conversion was not for the purposes of marriage.