I would willingly look a 9 year old child laborer in the eye each time I buy a cheaply made T-shirt, and I would feel good about myself.
Consider: most people in developing countries are able to choose which job they will do. It is true that some (far too many) are held in debt bondage, but the vast majority are not. Therefore, if the 9 year old child laborer is working to make cheap T-shirts, they're doing that because it's the best option available to them. If I refuse to purchase these T-shirts on principle, I'm denying them an income. I would much rather deny the income of first-world manufacturers who don't need my money.
Markets "decide" what is best by responding to price signals. If you buy a cheap T-shirt then you are saying to the market that you value low price over an ethical supply chain.
Buyers from first world countries do not (in general) deal first-hand with the labourers, they deal with factory owners who have the capital equipment necessary to do the manufacturing at the required scale. All the while that there is price pressure from the market (from you), the buyers will be pushing that price pressure onto the factory owners. As long as there is one factory owner who is willing to take that low price then all factory owners who want the same (or similar) contracts have to take that price, otherwise they are out-competed in the free market. They, in turn, push that price pressure onto their workers, preventing increases in wages.
The rest of what you said is correct: the workers take the jobs not because they want to work for such low wages, but because they are the best wages they can obtain given their skill levels. Buying cheap clothes almost directly contributes to the continuation of poverty.
As an aside, you mentioned child labour. A not uncommon strand to discussions on these issues is that people should "act more responsibly" and "choose" to get educated when they are young rather than waste their time for the short-term benefits of working. People send their children to work because they can't bring in enough money themselves, for reasons like disability (there are a lot more workplace accidents in countries with poor labour protections!) or plain poverty. How does the hypothetical 9 year old improve their lot if the best thing they can do at the age of 9 is get a job manufacturing clothes?
Let me respond to your final paragraph first. I actually don't agree with the idea that people should "act more responsibly" and "choose" to get educated. Or rather, I think they are never given the choice. Children working are forced to do so by the poverty of the parent, who were in turn forced into their life path by poverty. Of course this is not always the case, but what I have seen indicates that economic poverty is primarily due to a poverty of choices. Economic poverty flows from that.
That said, I stand by my belief that the free market is an effective way (and perhaps the most effective way) to lift people out of poverty. It's cruel, slow, and painful, but it also produces long-term improvements at a scale that mere charity cannot match. I think targeted aid is great, and I expect that it would be far more effective on a per-dollar basis, but cheap labor has a greater impact simply because the world economy throws a lot of money at it.
I disagree with your assessment that "buying cheap clothes almost directly contributes to the continuation of poverty" (However, see my caveat below). I just had a chat about this with my flatmate, who is an economist and has a much better grasp of the topic than I. He pointed out that, historically, sweatshops are only the first stage of economic development. Sweatshops provide a better (but still bad) income to unskilled laborers, which gives them enough free income to invest in their own human capital. Greater human capital attracts other investors, and the country pulls itself up by the bootstraps. South Korea, Taiwan, and Japan have built developed economies by starting with sweatshops. China is in the process of doing this. In contrast, India has more restrictions against foreign investment and sweatshops, and their economic growth has lagged behind countries with less restrictive laws.
Or, to put it more simply: sweatshops are an important stage on the road to a developed economy, because initially cheap labor is the only thing that an undeveloped country can offer to investors. If you don't allow sweatshops, you don't get any investment, and the development process stops before it truly started. There's a lot of inefficiencies, and a lot of greedy people (including us as consumers) taking bites out of the pie, but it works better than anything else we've tried.
As I mentioned above, my position is tempered by an important caveat. You point out that some countries have an ethical supply chain. I initially discarded this based on the assumption that 'ethical' usually just means 'first world', but I realize now that you are correct. I haven't yet come across a garment manufacturer with a truly ethical supply chain, but that doesn't mean they don't exist. I will make it a priority to search out such a supplier, and provide them with my patronage. If you have any suggestions, I would be happy to hear them.
You've said quite a lot, so please forgive me in advance for any lack of coherence.
I would be slightly wary of making predictions from historical trends, because we don't have a very large sample size, and there are a lot of confounding factors, a good example being cultural differences between countries, by which I mean differences in social organisation rather than what clothes people wear.
Irrespective of those differences, the general idea of an increase in skill level with continued investment has both common sense and, as far as I am aware, theoretical appeal. The issue, however, is really one of degree, and that is important for people at the hard edge of poverty. Whether it is going to take 1, 10, or 100 years for the conditions in a given region to improve matters to people who have so little money to live on that they can't afford to send their children to school, or pay for proper housing. If there is no money left over for workers to improve their lot because they are living at subsistence level, which is distressingly common, then the development process you describe cannot take place.
Distorting the free market in order to temporarily redistribute wealth downwards, with the goal of speeding up the process of improvement, is in my view an ethical action. An example of the kind of distortion that I think could be appropriate is supporting an oversight organisation along the lines of the Forest Stewardship Council. Bodies like that are by no means perfect, and are themselves often subject to market pressures, however they seem to me to be a pretty good balance of economic freedom and social ethics.
A slight aside; it strikes me that a not insignificant proportion of the increase in wealth that we enjoy has come not from people being able to afford more of everything, but rather from technological progress in methods of producing things. So for example the desk on which my computer sits is a fairly cheap affair constructed from what looks to me like veneered MDF. I do not have a nice oak writing desk, because those are still very expensive. The same kind of comparison applies to almost everything I own that has a historical counterpart: food, clothing, furniture, writing materials.
My point is that my increased "apparent" wealth seems to be driven as much by these technical improvements as by an increase in "actual" wealth (I wouldn't dare guess the actual ratio beyond saying that I feel the factors are comparable). Furthermore, I am confident in saying that technological improvement is not predicated on market freedom on the basis that there have, historically, been civilisations who have undergone technological improvement without free markets: the Romans being one example amongst many. I would also point out that we don't have enough data to know whether the current burst of technological improvement was dependent in any way on market freedom, because we only really have one data point: this particular rise of standards.
To return to your question about an ethical supplier: it kind of depends on your taste. My partner is Indian and I seem to end up getting my clothes on visits there these days, mostly as a result of laziness whilst here. Fabindia is one supplier that seems to put quite a lot of effort into ensuring good conditions in their supply chain, though they don't have the range you'd expect from a western supplier (they do seem to ship internationally though, not sure about stores). There are lots of workers cooperatives in Kerala, where my partner is from, so all the stuff they make must get on to the market somehow! Also, I don't know if you're aware but somewhat coincidentally NPR's Planet Money are doing a series of podcasts and associated things on t-shirts, and the economics of the global t-shirt trade, that sounds like it's going to be quite interesting.
Consider: most people in developing countries are able to choose which job they will do. It is true that some (far too many) are held in debt bondage, but the vast majority are not. Therefore, if the 9 year old child laborer is working to make cheap T-shirts, they're doing that because it's the best option available to them. If I refuse to purchase these T-shirts on principle, I'm denying them an income. I would much rather deny the income of first-world manufacturers who don't need my money.