Perhaps someone else can offer another alternative, but the 2 options I see are:
1) Continue your water changes normally, but switch to a conditioner that also binds ammonia. Prime is a good one, but there are others. This will chemically "lock" the ammonia in your tap water so that it does not harm your fish until your biological filter can take care of it.
2) Switch to using bottled water instead of tap water. This is inconvenient, expensive, and would not contain beneficial minerals like your tap water.
If you can't tell, I favor option 1.
