Fabarm makes a good one for around $1,000. To me, that's a cheap double.
I just don't have any experience with the Turkish guns. They look very nice, and they seem way better than the price would suggest. Most have forend iron fitted WAY too tight, triggers tend to suck, and of course, the "average" stock dimensions. If the gun is close enough for fit, my bet is the Turkish guns would be fine to learn with.
But Mag410 makes some really great points. Dollar for dollar, you'll never be able to match quality between a semi-auto and a double. For a grand you can get a "decent" double. For a grand you can get an excellent semi-auto. For two grand you're just getting into "good" for doubles, while you're squarely at world class for a semi-auto.
A used Remington 11-87 or 1100 is about as soft shooting as you're going to find. While they do require more maintenance than current top of the line semi-auto's, it's not by much. Both 1100 & 11-87 are VERY reliable shotguns; just clean them every 500 or so rounds (is that too much to ask?).
And the gun can grow with the shooter. Short stock & barrel when they're little, and just replace stock and barrel as they grow; the gun can last a lifetime.
I'm a Browning Auto 5 guy myself, but I won't be buying one of those for my daughter; I'll be buying her an 11-87 20ga.