How much will it cost me to move my septic system from the backyard to my side yard?

Without knowing what part of the country (and, indeed, which country) you are in, questions of costs are very difficult to answer. But I have seen estimates exceeding $10,000 easily for installing a new septic system, which is essentially what "moving" would entail. In addition to the new system installation, you would have to make sure that the old septic tank cannot collapse, for which it will need to be pumped empty ($500-$1000) and filled in with gravel.

If the septic tank itself stays, perhaps you may be able to do it less expensively. If you only move the drain field, you'll need a new pipe to the new distribution box and the perforated pipes for the field itself but at least there will be no excavation, large machinery on the property and other such headaches. I have a hard time picturing the move from back yard to the side of the house that only entails new pipes though. Not every property allows for such changes.

Do you have a mound drain or just a gravity fed flat field? I could not get it from your question? If you have a flat drain field, you may decrease the space requirement by installing a pump chamber for the effluent and a mound instead (I intentionally omit aesthetics here).

Anyhow, solutions do exist but I just wanted to mention that projects like these are expensive, can easily go wrong and should generally be avoided if at all possible.

I parting thought: check with your municipality, perhaps there's a sewer that was installed later than your house was built that you can connect to? That would be an ideal solution.