When all else fails, I make right turns until I find a safer route. I'm not allergic to taking a longer way 'round if it's safer. I'm rarely in such a hurry that I need to take a risk or stress out over potential danger.
This is among the trickiest intersections I encounter occasionally. It's not too bad now because Fort Worth has been emphasizing bike lanes and awareness (the mayor is an avid cyclist too - she sponsors and participates in rides). To navigate this starfish intersection I take the middle of the lane, preferably behind a car heading the same direction to break a path for me. So far, so good.
There's a more dangerous route heading eastward toward downtown, involving a dangerous merge with a highway on-ramp, competing for attention with a busy four-way intersection. I try to avoid it during rush hour. If it's rush hour or my legs aren't up to sprinting, I'll take a longer way around to avoid it.
And to my west, in an otherwise rural area, there's a tricky frontage road left turn across a bridge, with traffic merging from the highway in my blind spot. Seems unusually busy even on a Sunday. And the approach to the intersection is a long uphill grade, so my legs are usually nearly spent by the time I need to sprint. Tricky, but I've become less shy about taking the lane even if folks have to slow down a bit behind me. If they choose to live that far out in the country while commuting from the big city, what's another few seconds to wait for me to huff and puff across the intersection? That's what I'm planning to tell the first person to complain, assuming I can manage to speak coherently through the asthma attack and possible confrontation with the business end of a pistol.