Is picric acid a very strong acid?
Yes it is very strong, albeit not compared with Inorganic acids like HCl...

As far as organic acids are concerned, it is really one of the strongest ones... as its pKa is 0.3 [ more acidic than acetic, benzoic, formic, oxalic, trichloro-acetic acids...]

Possibility of intramolecular H-bonding, though exists, is offset reasonably as the negative charge of phenolic oxygen is very nicely delocalized over the 3 nitro groups. [In fact, if we compare the pKa's of phenol (9.98) with o-, m-, p- nitro phenols (7.23, 9.88, 7.15 respectively); we can see that H-bonding is not decreasing the acidic nature "upto greater extent"]

It is always possible that due to bulkier -NO2 groups, the repulsions betweens the LP of O atoms in -OH might be leading to the H of -OH group to be oriented far off from the range of effective H-bonding