"Feumaidh tu brògan."
Translation:You need shoes.
This is called the 'homorganic rule', 'dental rule' or the 'sgian dubh' rule. It is discussed on a couple of other questions, such as https://forum.duolingo.com/comment/35685936?comment_id=35871810
Basically, letters are less likely to lenite after sounds made in the same part of the mouth. But the rule is not applied consistently so you have to learn them on a case-by-case basis.
The first thing is that it is based on how it used to be pronounced. So never mind if we don't pronounce dh even resembling a d - if it looks like it has a d in it then that is good enough.
I think all the ones that affect thu are the three verb forms that end in -dh or -as
Feumaidh tu brògan 'You need shoes'
Dh'fheumadh tu brògan 'You would need shoes'
Ma dh’fheumas tu brògan 'If you need shoes'
and the forms of the copula which all end (or used to end) in -s, -n or -dh
Is tusa Iain 'You are Iain'
Cha tusa Iain ‘You are not Iain’
An tusa Iain? ‘Are you Iain?’
Bu tusa Iain 'You were Iain'
Bu does this because it used to be budh. Cha does it because it had an n at the end (as you know from the form chan). It seems to be just chance that all the forms of the copula do this – it just happens that each ended in a relevant letter.
Note that no one seems to know why thu is lenited anyway. There is no obvious reason and it isn't in Irish, as you say. But whatever the reason for lenition in the first place, this is the reason for delenition. D
Yes this is wrong because there is no 'your' in the Gaelic. This is not just being picky as it is different. Any shoes would do - they don't have to be yours. Perhaps some new ones or I could lend you some.
If perhaps you thought the tu meant 'your', no it meant 'you'. It means 'your' in Spanish.