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adjust how you play. I'm not sure what else to tell you. I've never had an issue and would probably have to see you in action to know what you are doing wrong. cause it's def you that's doing something incorrect.

What type of guitar and bridge? I find that with certain bridges (especially the BC Rich wraparound) if it's high up, it's incredibly uncomfortable to play on the low strings.

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Why would you want to have your pick in that kind of a position, as in the second picture?

We need pictures that show your posture in a larger scale. Those pics don't really show anything, except that your bridge isn't floating so it's alright to rest your palm on it if you need to.

Try a different strap length?

For the 6th string rest your palm just to the left of the pickup and pick just above it.
For the 5th and 4th strings just move your palm a little to the right, so that it's resting on the actual left side of the pickup (and muting the 6th string).
By 'rest your palm' I mean the left side of your palm, so the area below the thumb. The right side of your hand should be in the air. I don't know how to describe it any better.

That's how I do it with both floating and fixed bridges.

EDIT: Here's a very shitty webcam picture so you can sort of get the idea of how I do it. I don't normally raise my fingers like that, just did it to show where my palm is.