You are obviously very undermedicated with TSH of 45, yet FT4 of 5.7....but curious to know how your doctors confidently know you are a poor converter if they never seem to have done a FT3 test? It seems odd to me the 150 mcg levo( now upped to 200 mcg) has only produced a small amount of FT4 in the blood....have you got previous blood tests to share as your dose was increased from say 50 mcg? Perhaps you will finally need double the levo. dose. Have you had your folate, ferritin, b12 and vit D tested as high end of ranges needed for optimal thyroid health and good conversion, plus selenium etc? Are you known to have thyroid antibodies (TPO or TgAb), ie Hashimoto’s disease which can cause huge fluctuations in thyroid blood results? Hopefully other will comment. Meanwhile.....
pity the endo couldn’t have given you at least a limited trial T3. Many of us have not received any offer of T3, and get ours privately( at a reasonable price) and go DIY with help from forum members.
There is something very wrong to be taking 150mcg Levo and produce those results.
As Judith says, how can anyone say you don't convert T4 to T3 when they haven't tested FT3 - if that has come from a doctor then they know nothing about treating hypothyroidism, so run away and find a doctor who does.
T3 can be prescribed on a private prescription where a need is shown, so again you've been given incorrect information.
TSH and FT4 testing is not the complete picture. You need all the tests that Judith has mentioned:
TSH
FT4
FT3
Thyroid antibodies - Thyroid Peroxidase and Thyroglobulin
Vit D
B12
Folate
Ferritin
If you GP can't do them all (which is very likely) then take matters into your own hands to help yourself and do a private test with one of our recommended labs, you can do a fingerprick test or venous blood draw if preferred at extra cost. Either of these tests cover everything needed:
Medichecks Thyroid Check ULTRAVIT medichecks.com/thyroid-func... and you can use code THYROIDUK for 10% of the normal price
Blue Horizon Thyroid Check PLUS ELEVEN bluehorizonmedicals.co.uk/t... and use code FEBRUARY for 20% off the normal price (says today only but may be on until the end of the month).
Post results, with reference ranges, when you have them and members will be happy to offer guidance.
There is nothing to stop a Dr, whether private or NHS, from providing you with a private prescription for T3 if they believe it would be beneficial for you; and as a private Consultant he has no need to consider the views of the CCG. Similarly, there is nothing to stop an NHS Endo from recommending a trial of T3 within RMOC/NHS guidelines, if he thinks it necessary and that you would benefit from it. As you don't have a FT3 test result your GP can't possibly know what is your level of FT3 and therefore can't know whether you do or don't convert effectively. Given your high TSH, simply increasing your dose of Levo in the vague hope that something might improve, is inadequate. If it were me I would be asking for a referral to an NHS Endo in order to be better tested and diagnosed, if only to prove that you are one of the rare individuals that require a supraphysiological dose of T4 to be well; and if not, to confirm that you convert poorly and need a trial dose of T3.
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thyroid function test results 
Help understanding my thyroid test results.
Thyroid blood test results 
thyroid blood test results
