Bisected.
Requires mesa 32-bit.
Good commit is 20104f1926436e00171c8e64ca37fff9ffbd7096.

The next commit 314879f7fec07cedb5263681173a22d522a8ac9a breaks building with error "implicit declaration of function ‘__builtin_ia32_clflush’". That issue was fixed in BZ https://bugs.freedesktop.org/show_bug.cgi?id=101774 by commit 6b05c080f202c25531f59b547d5e6ba339cedd09.

Commits 314879f7fec07cedb5263681173a22d522a8ac9a + 6b05c080f202c25531f59b547d5e6ba339cedd09 fixes build but fails with:
"intel_batchbuffer.c:720: _intel_batchbuffer_flush_fence: Assertion `!brw->no_batch_wrap' failed"

Also applying of 6b05c080f202c25531f59b547d5e6ba339cedd09 to the last good commit 20104f1926436e00171c8e64ca37fff9ffbd7096 gives current issue.
So issue is combined in two commits: 6b05c080f202c25531f59b547d5e6ba339cedd09 and 314879f7fec07cedb5263681173a22d522a8ac9a

Proposed patch: https://patchwork.freedesktop.org/patch/247065/
Levente, are you able to check it (requires clean and rebuild of mesa)?

I wouldn't say I'm a newbie, but I don't have any experience in compiling Mesa for myself. I might look into it, but that will take a while.

Hi, I sent instructions to your mail. They are 100% tested and valid for ubuntu 16.04

Please fill free to ask and provide your feedback

With the help of Denis, I did some tests. Although I couldn't make Steam use the compiled libraries directly from the Mesa 18.3 development branch, I used the method of overwriting the i965_dri.so file with the one I compiled. It worked. So it seems that the bug was fixed somewhere along development. Before closing this report as solved, I would still like to wait some more time, just in case.

The fix landed upstream. Please check if this fixes the issue.

I tried it with padoka ppa (18.3 development version). The issue seems to be fixed, it works how it should be.