Problemet är tydligen större än vi trodde, men chefen tror att det blir bättre snart.

Because problemet is the definite form, meaning the problem.

• ett problem = a problem
• problemet = the problem

Problem is an ett noun, so its singular definite ending is -et.

Useful full set:

• ett problem = a problem
• problemet = the problem
• problem = problems
• problemen = the problems

Tydligen usually means apparently, evidently, or it seems.

It suggests that the speaker is not presenting the information as a simple direct fact, but as something that seems to be the case or has become clear from evidence.

So:

• Problemet är tydligen större...
  = The problem is apparently / evidently bigger...

It often has a slightly softer or more observational tone than a plain statement.

This is because Swedish main clauses follow the V2 rule: the finite verb normally comes in the second position.

Here:

• Problemet is first
• är must come second
• tydligen comes after the verb

So the order is:

• Problemet är tydligen större...

This is very typical Swedish word order. In English, adverbs like apparently can move around more freely, but Swedish is stricter in main clauses.

Större is the comparative form of stor, meaning bigger, larger, or greater.

This adjective is irregular:

• stor = big
• större = bigger
• störst = biggest

So:

• större än vi trodde
  = bigger than we thought

Also, större does not have to mean physically larger. It can also mean more serious, more extensive, or greater in scale.

Än means than in comparisons.

The pattern is:

• comparative adjective + än

Examples:

• större än = bigger than
• bättre än = better than
• snabbare än = faster than

So:

• större än vi trodde
  = bigger than we thought

After än, Swedish can use:

• just a noun: större än problemet innan
• or a full clause: större än vi trodde

Because it refers to what we thought earlier.

• vi trodde = we thought
• vi tror = we think

In the sentence, the idea is that our earlier belief was one thing, but now we know the situation is worse:

• Problemet är tydligen större än vi trodde
  = The problem is apparently bigger than we thought

This works very much like English.

No. This is a very important false friend.

In Swedish:

• chef = boss, manager, head
• chefen = the boss / the manager

The English word chef in the sense of professional cook is usually kock in Swedish.

So here:

• chefen tror...
  = the boss thinks...

not the cook thinks...

Att means that and introduces a subordinate clause.

So:

• chefen tror att det blir bättre snart
  = the boss thinks that it will get better soon

In English, that is often omitted:

• The boss thinks it will get better soon

Swedish can also sometimes omit att in speech, but using att is completely normal and often clearer.

Here det works like a dummy subject, similar to English it in it gets better.

It does not necessarily refer to one specific thing. Swedish clauses usually need a subject, so det fills that role.

So:

• det blir bättre
  = it gets better / it will get better

You can think of det here as grammatical rather than strongly meaningful.

Because Swedish often uses the present tense to talk about the future when the time is already clear from context.

Here, the word snart (soon) makes the future meaning obvious:

• det blir bättre snart
  literally: it becomes better soon
• natural English: it will get better soon

This is very common in Swedish.

Compare:

• Jag ringer imorgon. = I’ll call tomorrow.
• Vi ses snart. = We’ll see each other soon.

So present tense in Swedish often covers near-future meaning.

Because att det blir bättre snart is a subordinate clause, and subordinate clauses follow normal subject-verb order.

Here:

• det = subject
• blir = finite verb
• bättre = complement
• snart = time adverb

So the order is straightforward:

• att det blir bättre snart

A useful general point:

• Main clauses in Swedish follow V2
• Subordinate clauses introduced by words like att do not use that same main-clause pattern

Because adverbs of time often come late in the clause in neutral Swedish word order.

So:

• det blir bättre snart

is the most natural, neutral way to say it.

You can move snart in some contexts for emphasis, but the end position is very common and natural here.

Yes, but the original sentence is very natural.

A few possible variations are:

• Problemet är tydligen större än vi hade trott...
  This can sound a bit more explicit about the earlier belief: than we had thought

• ...men chefen tror att det snart blir bättre.
  Also possible; this puts a bit more focus on soon

• Problemet verkar vara större än vi trodde...
  = The problem seems to be bigger than we thought
  This is slightly different in tone from tydligen

But the original sentence is idiomatic and completely normal Swedish.