Conditional type (unreal past)

1. my friend will have a party tomorrow, i think she'll invite me:
if she invites me, i go/i'll go
beni davet ederse giderim/gideceğim

2. party time gets closer and she's not invited me yet. another friend asks me whether i'll go, i say:
if she invited me, i'd go (but she didn't, so i don't/won't go)
beni davet etseydi/etmiş olsaydı giderdim (ama davet etmedi, ben de gitmiyorum/gitmeyeceğim)

3. the party is over, she hadn't invited me and i didn't go:
if she had invited me, i'd have gone
beni davet etseydi/etmiş olsaydı giderdim

you see, we use the same structure for both 2nd and 3rd types. we could say in the 3rd case,

beni davet etmiş olsaydı gitmiş olurdum (exact translation)

it's grammatically correct but very rare, a bit clumsy and it doesn't add to the sentence any more sense.

your sentence would be translated as,
if i had seen her, she wouldn't have left
onu görseydim/görmüş olsaydım ayrılmazdı (terk etmezdi/bırakıp gitmezdi).

bonus:
i wouldn't have gone even if she had invited me.
beni davet etseydi/etmiş olsaydı bile gitmezdim.

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let me/you etc drink:

içeyim (1st person singular)
içesin (2nd person singular)
içe (3rd person singular)
içelim (1st person plural)
içesiniz (2nd person plural)
içeler (3rd person plural)

remarks:

1. in folk songs -yi/-yı of the 1st person singular often drops:
kalayım -> kalam: let me stay
bakayım -> bakam: let ne look

öleyim -> ölem: let me die
göreyim -> görem: let me see

2. translation may differ depending on the context:

bırak seveyim
rahat edeyim
git de gebereyim (*)
(nazan öncel)

let me love
let me relax
tell me to go and i die (or, go and i die)

anan öle cemil
baban öle cemil
yetim kalasın cemil
benim olasın cemil
(a folk song)

may your mother die, cemil (a male name from arabic, meaning beautiful)
may your father die, cemil
may you become orphan, cemil
may you be mine, cemil
(here the structure means a malediction or prayer)

bu belalı başınan ben nere gidem
bu belalı başınan kime ne diyem (**)
(enver gökçe/ahmet kaya)

where do i go with this head full of troubles
what do say to whom with this head full of troubles

3. içesin, içe, içesiniz and içeler are rarely used in spoken language.

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(*) gebermek: to die. use this word with extreme caution (and i recommend not to use it until you really feel yourself familiar with the situations it can be used): it reflects strong feelings of hatred, despise or anger about the person that died. use öldü or vefat etti (deceased, an euphemism, use this if the person is dear to you or to the person you talk to) safely. in the song nazan öncel uses it for emphasis. i don't know any english word that corresponds to it.

(**) başınan = başla (ie, baş ile): with head. gidem -> gideyim, diyem -> diyeyim.