How do I deal with 'compostable' bags?

TL;DR: Generally it's best not to add compostable plastics to your compost pile because most plastic types do not break down in home compost piles. If the bag has a genuine 'compostable' logo then give it 90 days to see if this bag is compostable in your pile or not.

The problem with compostable plastics is that many plastic manufacturers use a different definition of compostable than you'd expect.

That which is capable of undergoing biological decomposition in a compost site such that the material is not visually distinguishable and breaks down into carbon dioxide, water, inorganic compounds and biomass at a rate consistent with known compostable materials. (source: ASTM D6002 standard)

This definition says nothing about the needed temperatures in the composting process nor does it say anything about the toxicity of the resulting substances. The only criterium is the speed at which the plastic breaks down. Luckily there are stricter and better standards; the US ASTM D6400, the European EN 13432 and the Australian AS4736. These standards have a very low limit for the amount of heavy metals and toxic substances that are allowed in the resulting compost (but this limit isn't zero).

The result of all these different standards is that most compostable plastics can be broken down, but usually only in industrial recycling centers where the plastics are heated to very high temperatures for several days. If you put compostable plastics on your compost heap you'll find that it won't break down because the temperatures aren't high enough. The resulting 'compost' from industrial recycling centers usually is fairly safe, but it depends on how strict the definition of compostable is where you live and which standard applies.

The only standard that guarantees that a product can be home composted is the Australian AS 5810. There are also some marks that guarantee this, but so far I haven't seen them in real life so they are probably scarce. More info about this in this question about home compostable plastics.

Note that there is a difference between compostable and biodegradable plastics. For biodegradable plastics there are no maximum levels of toxic chemicals or heavy metals, they just need to break down much faster than traditional plastics.

More info:

EDIT: I did some research; here it says that the particular bags you are referring to are ASTM-D6400 certified. This means that 60% of the material should break down within 180 days. The question remains under which circumstances. There is one review that says that

... they break down in my compost 30-60 days after mixing in depending on the temperature.

so these bags may be really compostable in your compost pile. I'd suggest you give it 90 days to see if this is the case.