If so you're in the golfing majority. The pull slice with the longer clubs is the most common shot in golf. Your full shots with the short irons probably start left as well, but tend to stay in that direction with no curve to the right.
Understanding swing path is the first step in correcting this problem. Because we stand to the side of the ball the club must swing on an in-to-in path much like a saloon door on a hinge. The club moves from the inside on the downswing and then back to the inside during the follow-through.
Shots starting to the left of your target indicate that your clubhead is not reaching the ball until it has begun returning to the inside. In other words, you are contacting the ball too late in your swing. This occurs if you are playing the ball too far forward, or to the left in relationship to yourself.
Your ball curving to the right is further proof of a too far forward ball position because it forces you to address it with your body and shoulders aligned too far to the left of the target. This alignment makes you grip the club with your hands turned too far to the left. This grip position leaves the club face open (facing right of its path)at impact creating a slice.
To correct this play the ball farther back (to the right) in your stance. This will align your body and shoulders more to the right and your hands will want to turn more clock-wise on the grip. You are now in a position that will allow you to swing the club on an in-to-in rather than an out-to-in swing path. With your hands turned farther to the right you will also now be able to square the club face to its path at impact.
If your shots now begin to the right of target you have moved the ball too far back and you're reaching the ball too early in the swing. Simply move the ball a little more to the left until it starts straight toward the target. If your shots start on target but still curve to the right turn your hands a little further to the right.
Correcting your ball position, body alignment and grip will not only move you from the majority to the minority of golfers, but also help you hit longer, straighter golf shots.