I get a lot of reader questions about front bags, and particularly about whether I've found a good way to carry a bag on the handlebars without using additional support. Short answer: In my experience, it depends on the bike, on how much weight is being carried, and on how the bag is attached to the bars. These factors interact with varied results.
On roadbikes, I have tried carrying unsupported loads on the handlebars using several methods. Bags attached with straps are appealing, because they do not require permanent hardware installation. However, on longer and/or faster rides I find it difficult to keep them from sliding along the handlebars, no matter how tight the straps and how light the load. If the bag slides when I lean on a turn at high speed, it feels distracting and disconcerting, even if it is too light to affect steering.
A few manufacturers have come up with hardware to address this, including the Nitto Bar Sack Rack, the Brooks Cornwall system, and the Rixen & Kaul KlickFix adapter. The latter I've had a chance to investigate on Pamela Blalock's bikes - who uses it on most of her roadbikes.
The KlickFix adapter is mounted on the handlebars and remains there when the bag is detached. It is compatible with many bags from different manufacturers, the most popular being the waterproof Ortlieb bags, available in a variety of sizes. This is a very stable system for carrying unsupported weight on the handlebars, and it worked for me with light loads: I felt zero movement. However, when I added my DSLR camera (between 2 and 3lb with lens), I found that I had poor control of the steering. I reported this to the bag's owner, who confirmed that she does not carry that much weight in a handlebar bag on this or on any of her other roadbikes; she uses this system to store lightweight items only.
Of course, it is not clear whether having the weight lower and supported by a rear rack would make any difference on the specific bike I rode. Anecdotal evidence suggests that it very well could. On the other hand, I have seen touring bikes with large, heavily laden bags attached to the bars using this same method, ridden successfully. So the bike does play a role, as, no doubt, does rider sensitivity to high/ unsupported loads.
With upright city bikes - especially heavily built ones - there is generally more tolerance. I have ridden such bikes with all manner of baskets and bags attached to the handlebars without lower support, and for short distances it's been fine if the weight is only a few pounds. I've even managed to attach my Po Campo pannier to the handebars of a Raleigh DL-1, and ride with this setup with 5lb+ of weight in the bag. The bag does slide side to side and affects steering a bit, but since I am not riding fast or cornering aggressively, it doesn't bother me. The heavier the load, the more likely it is to become problematic. And then again, there are those who simply hang heavy shopping bags off their handlebars and blithely pedal away.
The general consensus is that carrying weight high on the handlebars and without additional lower support is not ideal. A stable system such as the R&K KlickFix adaptor is probably your best bet, but still there are limits to how much weight will feel comfortable. If you have a success story, do share. Personally, in the absence of a front rack, I prefer to carry weight at the rear of the bike.