After painting my Leman Russ mostly by brush, I can say a few things that help with streaking...
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1. Yes, large brush is better. Some of the streaks you get will be from the side of the brush carving a path along the miniature. You can reduce that streaking by swiping back across areas rather than just doing a single stroke, but none the less using a larger brush reduces those streaks. But don't go too overboard, if the brush is larger than the details then you'll struggle to lay down the paint neatly without obscuring detail (though I found using a large brush in circular motions over detailed areas, just going round and round in circles both clockwise and anticlockwise managed to stop the paint gathering where I didn't want and spread it evenly over detailed areas).
2. Use a soft brush. If you're getting streaks from the brush itself, it might be that your brush isn't soft enough. I have no idea what the
GW tank brush is like, I've never used it, but if it's not a very soft brush or if you've used it for drybrushing (which tends to make the bristles go a bit stiffer over time) then that might be a problem. I used a couple of big brushes I bought from an art store and I straight away noticed the one I used more often was leaving streaks because the bristles were stiffer than the one I hadn't used as much.
3. Water down the paint to a good level. Too watery and you might have problems other people have mentioned above, but having a nice thin paint allows it to self smooth to a certain extent. So play with the consistency.
4. Multiple thin layers. You WILL get brush strokes within each layer, the best you can do is make sure the paint is thin enough so that it's not actually creating a texture, so the next layer will hide the strokes from the previous layer. For my Leman Russ I found I was doing 3 to 4 layers over most of it to get a solid coat while remaining smooth.
5. Don't use too much paint on your brush. It's very tempting when you have large areas to just use lots of paint to get it finished quicker, but you'll also be more likely to leave streaks.
I'll probably think of other things later, but that's what comes to mind. I also suggest going for a spray. I painted my Leman Russ by hand because I couldn't find sprays that matched my needs and didn't want to buy an airbrush because I was going overseas, but it took ages to lay down solid coats of the grey I was using so I'm definitely looking at buying an airbrush when I get back home rather than brush painting the next Leman Russ.