That's the classic OOR (Out-Of-Round) problem. Ultimate solution, go to what is called "Mandrel Less" or "Center To Center" turning. (Lots of good posts on those subjects).
The usual suspects in this are running the tail-stock in too tight, not having the mandrel centered to start with, or too much pressure while turning. My technique is to put the mandrel in the head stock, but leave the brass disk loose at first. Rotate it a few times by hand with the end of the tail stock just off the end of the mandrel. If it looks like it's pretty well centered, keep turning the headstock by hand, and slowly turn in the live center until it turns along with the mandrel. STOP.... Don't crank it in another little bit, really, it's not needed. Now you can tighten up the brass wheel, again, just tight enough to keep the blanks from rotating when you turn them. I actually check it a few times, I need it tighter for roughing them out, then once it gets round I can usually ease off the tension a touch.
Don't use too much pressure, that will tend to make the mandrel flex.
It's also possible that you have some bushings that are OOR to start with. It's not unheard of by any means. There are some good folks here who I'm sure will offer more and better suggestions, but if you don't over-pressure things it will probably go a long way to help with the problem.
BTW, Mandrels are cheap, I HIGHLY recommend getting a few extras for the bench. Many of us here have managed to turn on the lathe without getting the tailstock up against the end of the mandrel, and it tends to get real "Exciting" then.
One Last Thing (as Columbo would say), are you using a true 60 degree live center in your tail-stock, or is it the multi-purpose one that came with the lathe. Those aren't a true 60 Degrees, and can allow the end of the mandrel to move a bit as you turn.
Hope this helps,
Steve