Hata yapsak bile birlikte düzeltebileceğimizi biliyorum.

Yes.

• Hata – mistake / an error (no article in Turkish; context gives a or the)
• yapsak – if we (were to) make/do
  
  • yap- – root: to do / to make
  • -sa- – conditional suffix (if)
  • -k – we (1st person plural ending in conditional forms)
• bile – even (if), even though
• birlikte – together
• düzeltebileceğimizi – that we will be able to fix (it)
  
  • düzelt- – root: to correct / to fix
  • -ebil- – can / be able to (potential ability)
  • -ecek- – future tense (will)
  • -imiz → -ğimiz – our / we (1st person plural; here marks we as the subject of fix)
  • -i – accusative case marker; shows the whole clause is the object of know
• biliyorum – I know
  
  • biliyor- – know (present, ongoing)
  • -um – I (1st person singular)

So literally:
Hata yapsak bile birlikte düzeltebileceğimizi biliyorum.
→ Even if we make a mistake, I know our-being-able-to-fix-(it)-together.

Both are possible, but they feel slightly different:

• yaparsak (yap-ar-sak)

  • yap- – do/make
  • -ar- – aorist (do in general / when we do)
  • -sak – if we
    → if we make (a mistake) in a more straightforward, factual way.

• yapsak (yap-sa-k)

  • yap- – do/make
  • -sa- – conditional
  • -k – we
    → often sounds more hypothetical, softer, like if we (happen to) make / even if we were to make.

In Hata yapsak bile, yapsak fits the idea of a possible, hypothetical mistake and pairs very naturally with bile (even if).
Hata yaparsak bile is also correct and common; it sounds a bit more like in the case that we do make a mistake.

bile means even in the sense of even if / even though.

In Hata yapsak bile:

• literally: If we make a mistake even…
• functionally: Even if we make a mistake…

Position:

• Most naturally: Hata yapsak bile (as in the sentence)
• You can also say: Hata bile yapsak, birlikte…
  
  • Now it’s closer to Even if we (actually) make a mistake… with extra emphasis on hata.

You usually place bile after the thing you want to emphasize as “even that”:

• Biz bile hata yapsak… – Even we, if we make a mistake…
• Birlikte bile düzeltebiliriz. – We can fix it even together (even under the condition of togetherness) – sounds odd here, but shows the pattern.

In this sentence, bile naturally comes after the verb phrase yapsak.

In Turkish, the subject pronoun is usually inside the verb ending, so it’s often dropped in speech and writing.

• yapsak = if we (were to) do/make
  
  • -k marks “we”.
    So biz (we) is understood and not needed.

If you want to emphasize we, you can add it:

• Biz hata yapsak bile, birlikte düzeltebileceğimizi biliyorum.
  → Even if *we make a mistake, I know that we will be able to fix it together.*

Turkish often omits bir (a/an/one) when talking about something in a general or generic way.

• Hata yapsak bile…
  → literally “If (we) make mistake even…”
  → understood as Even if we make a mistake or Even when we make mistakes (generic).

If you say bir hata yapsak bile, you’re pointing more specifically to a single instance:

• Bir hata yapsak bile, birlikte düzeltebileceğimizi biliyorum.
  → Even if we make a (single) mistake, I know we can fix it together.

Both are correct; the version without bir is more general, and very natural here.

birlikte and beraber both mean together, and in most everyday contexts they can replace each other.

• birlikte – slightly more neutral/literary/formal; used a lot in writing.
• beraber – slightly more colloquial/spoken.

In this sentence:

• Hata yapsak bile birlikte düzeltebileceğimizi biliyorum.
• Hata yapsak bile beraber düzeltebileceğimizi biliyorum.

Both are correct and natural. The meaning is essentially the same.

Yes.

düzeltebileceğimizi (that we will be able to fix it) comes from düzeltmek (to fix/correct):

1. düzelt- – root: to correct / to fix
2. -ebil- – potential suffix: can / be able to
   
   • düzelt-ebil- → can fix
3. -ecek- – future tense: will
   
   • düzeltebil-ecek- → will be able to fix
4. -imiz → -ğimiz – 1st person plural (our / we)
   
   • attached to a verb stem like this, it marks “we” as the subject of the embedded clause:
   • düzeltebileceğimiz → that we will be able to fix
5. -i – accusative marker
   
   • düzeltebileceğimizi
   • makes the whole thing the object of biliyorum.

So structurally:

• düzeltebileceğimiz – what we will be able to fix / the fact that we will be able to fix (it)
• düzeltebileceğimizi biliyorum – I know that we will be able to fix (it).

The final -i is the accusative case ending, marking a definite direct object.

The verb bilmek (to know) usually takes a clause as its object, and that clause is treated like a thing in the accusative:

• Onu biliyorum. – I know it / I know him/her.
• Gel-diğ-in-i biliyorum. – I know that you came.
• Düzeltebileceğ-im-iz-i biliyorum. – I know that we will be able to fix (it).

So -i shows that düzeltebileceğimizi is the “something” that is known.

Inside düzeltebileceğimizi, the -imiz (which surfaces as -ğimiz because of consonant softening) marks 1st person plural as the subject of that embedded verb:

• düzeltebileceğ-im – that I will be able to fix (1st singular)
• düzeltebileceğ-in – that you (sg) will be able to fix
• düzeltebileceğ-i – that he/she will be able to fix
• düzeltebileceğ-imiz – that we will be able to fix
• düzeltebileceğ-iniz – that you (pl/formal) will be able to fix
• düzeltebileceğ-leri – that they will be able to fix

Then you add -i for accusative:
→ düzeltebileceğ-imiz-i → düzeltebileceğimizi

Different structures:

1. Birlikte düzeltebiliriz.

   • We can fix it together.
   • Simple statement about present/general ability.

2. Birlikte düzeltebileceğimizi biliyorum.

   • I know that we will be able to fix it together.
   • Adds:
     • future nuance with -ecek- (will be able to),
     • knowledge with biliyorum, and
     • a full object clause (“that-clause”).

The original sentence emphasizes the speaker’s confidence about future ability:

• not just We can fix it, but
• I know that we will be able to fix it (when/if the time comes).

Turkish uses two main present-like forms of bilmek:

• biliyorum – with -iyor, formally the “present continuous”, but for stative verbs like bilmek, it’s the normal way to say I know (now).
• bilirim – aorist; can mean I know (in general), I tend to know, or I would know, and can sound less immediate or more generic.

In everyday speech:

• Biliyorum. – I know. (natural, neutral)
• Bilirim. – I (do) know / I would know (more generic or slightly emphatic, sometimes used in forecasts or habitual statements).

So in this sentence, biliyorum is the natural choice to express current, firm knowledge.

They are two different systems:

1. Finite verb endings (for main verbs and some subordinates):

   • yap-tık – we did
   • yap-acak-sak – if we will do
   • yap-sa-k – if we do / if we were to do
   • Here -k marks we on a conjugated verb.

2. Nominal/possessive endings on verb stems turned into nouns (verbals) like -dik, -ecek, etc.

   • yap-acağ-ım – what I will do / that I will do
   • yap-acağ-ımız – what we will do / that we will do
   • düzeltebileceğ-imiz – that we will be able to fix

So:

• yapsak – a finite conditional verb with -k = if we do/make
• düzeltebileceğimiz – a verbal noun (clause), with -imiz as the “subject” of that clause.

Two different layers of grammar, both encoding “we” in different ways.

Yes, that is correct and natural:

• Hata yapsak bile, biliyorum ki birlikte düzeltebileceğiz.
  → Even if we make a mistake, I know that we will be able to fix it together.

Differences:

1. Structure of the “that-clause”:

   • Original: düzeltebileceğimizi biliyorum.
     
     • uses a nominalized verb (-ecek + person + accusative) as the object.
   • Alternative: biliyorum ki birlikte düzeltebileceğiz.
     
     • uses ki to introduce a full finite clause: I know that we will be able to fix it together.

2. Style:

   • without ki (original): more compact, more typically Turkish in style.
   • with ki: also natural; sometimes adds a slight emphasis such as I do know that… and feels a bit closer to English structure.

Both are good; the original is more purely “Turkish” in terms of native-style embedding.

bile and da/de can both give an “even if / even though” sense, but with different flavors:

• Hata yapsak bile… – literally even if we make a mistake…

  • bile focuses more on unexpectedness / extremeness: even in that case.

• Hata yapsak da… – literally even if we make a mistake / although we make a mistake…

  • -sa da structure is more like “although / even if”, often slightly less emphatic than bile.

In many everyday contexts they overlap, and both would work here:

• Hata yapsak bile / hata yapsak da, birlikte düzeltebileceğimizi biliyorum.

The original with bile puts a bit more stress on even in the case that we make a mistake.