First, blowout is usually used to describe an explosive loss of air - BANG! In almost all those circumstances the tube has to escape the confines of the tire, and the resulting hole will be star-shaped or a long split in the tube. If that is what you are getting you are catching part of the tube underneath the tire bead. After a while the tube will push the bead up enough for the tube to briefly escape. After the blowout the tube often is back inside the tire.

If you are merely getting holes next to the valve that is not a blowout. It may be due to the tube having to fill up too much space next to the valve where it goes through the rim. As FBinNY has described it in detail before I found one of his posts about the subject.

This applies mainly to narrow rims, most often when combined with wide tires. A key factor is the inside width of the rim and the thickness (not width) of the tire at the bead. Thick beads on narrow rims leave only a narrow gap between them inside the rim, so the air chamber is sort of hourglass shaped with a circular section in the tire, the pinch, and another space below it within the rim.

When you inflate the tube, it first fills the space inside the tire, then the narrow section at the bottom blows down past the pinch filling the area in the rim. That means that this section of the tube may stretch 2-4 times more than the rest of the tube, hyperstretching it and often splitting or tearing it. You'll often see the results of this hyperstretching along the underside of the tube.

There's no easy fix, but these have been known to help.

Use the largest tube that fits, so there's more room to stretch
Rub talc on the inside of the tire, and outside of the tube, to allow the tube to shift more easily.
Inflate the tire only to a few psi and massage it to try to let the tube shift down.
Inflate the tube outside the tire to about double it's size (look for quality tubes that expand evenly, rather than crappy ones that inflate in small sections the way long balloon do). Let the tube stay this way overnight to relax the material a bit.
Inflate part way and leave the wheel alone for a while before coming to full pressure.
Hand stretch the tube, as many do with balloons to relax the material.
Warm the tube slightly (in the sun, or under a heat lamp) to make it stretchier.

I make no claims to precise science here, but I've helped lots of friends with similar issues, and with one or more of the above all have managed to put this behind them. Some may only be voodoo, but if it solves the problem I don't worry about exactly why or how. Part of me believes that the biggest factor is the tubes overall quality, and the rest may be like banging sticks together to keep the lions away.