Bridging is not required by code when joists are 12" and under. I would therefore conclude it's safe. Many claim adding bridging "stiffens" the floor but personally am not in that crowd.
Is the floor at all squishy?
It makes a huge difference by keeping the joist from tilting over when a load is applied. It also spreads load to joist adjacent joist The difference when added to an old house without it is tremendous, either blocking or bridging. I have seen two floors where adding it stiffened the floor up enough be be below the deflection (L/360) for tile flooring.
As you can see, there is no bridging between the second-floor joists and the floor above is a double layer of plywood. Should I be concerned? Should I add bridging? Or maybe something like a purlin running perpendicular to the joists? Or do nothing?
I don't know how new your build is or your what local code has been adopted but the IRC requires blocking at both ends of the joist. I am going to assume its been done on the perimeter by a rim joist but its hasn't been done over the support header. Now that may have been code when constructed so it doesn't necessarily mean you have to bring it up to code now
It specifies full depth 2" nominal thickness blocking at the ends not bridging.
R502.7
If bridging is required by the 12" depth, a 1x3 continuous strip across the bottom side is an acceptable solution R502.7.1
