are you tapping all the way through? From the photo, it looks like you have 4 threads that are tapered. For that thick of steel, I would get a longer taper. McMaster sells them with up to 7 to 10 threads of taper. I have a feeling this would make it easier.
If you're tapping bottomed holes, I would go as far as you can with the longest taper you have, then run a shorter taper, and then finish with a bottoming tap. I know that is expensive and time consuming to use 3 different taps per hole, but when hand tapping thick metal, I try to give myself every advantage I can get.
On a slightly related note, last week enlarged some 3/8"NPT and 1/2"NPT water-jacket fittings on some aluminum engine heads (I like my plugs and fittings to have as many threads of engagement as I can get, and the factory tap was only giving me about 3 threads). The heads were already torqued down to the motor, and I didn't want to re-do all of that.
To keep the shavings from falling down into the water jackets, I loaded the tap up with thick engine assembly grease (almost a paste, thicker than normal grease). It amazed me how easy the tap went through when it was loaded up with grease, compared to just putting a little tapping fluid on the threads when I was enlarging the holes on the intake manifold (which wasn't installed yet, so I wasn't concerned with the chips falling in).
I wonder if something like this might help you out? Maybe there's not enough lubricant in there, or it's dripping down through the hole? Maybe there's a thicker grease-like tapping lubricant?