Now I am very confused. I thought I knew that many very preterm babies had low serum thyroxine levels (this study confirms that). It seems to be a variant of sick euthyroid syndrome, levels tend to fall for about the first week of life then start to increase again. I thought it was well established that infants with a low thyroxine had poorer outcomes, in terms of long term impairments. To my mind it was not clear whether this association was causative, and randomized replacement trials have been generally negative, with no benefit demonstrated, but further appropriate trials were warranted.
Now a new study (Scratch SE, Hunt RW, Thompson DK, Ahmadzai ZM, Doyle LW, Inder TE, et al. Free Thyroxine Levels After Very Preterm Birth and Neurodevelopmental Outcomes at Age 7 Years. Pediatrics. 2014) suggests the opposite. In this group of babies, from the incredibly productive group in Melbourne, those babies that had higher thyroxine levels had worse outcomes. The main difference between this study and the previous ones is that the infants were studied at 7 years, so such things as language development can be examined in more detail. The babies were probably less sick than in some of the older studies, as very preterm babies are generally less sick than they used to be, and many of the older studies had less antenatal steroid use, and no surfactant. But I am still a bit confused by the reversal of the finding.
The authors note that recent models suggest that the sick euthyroid syndrome may well be adaptive and beneficial, rather than dangerous. If this is true, and higher T4 is a sign of poorer adaptation, then why the older studies showed the contrary remains a puzzle.
The authors also note that the method they used those several years ago to measure T4 is no longer considered optimal (and may be affected by low serum proteins which are common in the preterm). It would be interesting to have a new study, using the best modern assays, with cohorts from Australia, where high T4 seems to be bad for you, and from the US and Holland where low T4 seems to be the problem!
Neonatal Research 
Would a simple test for TSH at regular intervals from birth – preterm or normal – provide the information that would help you anwer your questions?
The answer is not really. I don’t usually approve comments from commercial sources, but this comment was on topic! If it became clear that it was important to do something about the transient hypothyroidism of the preterm, then a rapid accurate and low blood volume test might be valuable. All these babies are hospitalized, so any test would have to be approved by all the relevant agencies, and you would have to show some advantage over the standard lab methods.
For preterrms we are still at a phase of not knowing what to do about the findings!