Jeg synes, at den nye app er mindre besværlig end den gamle hjemmeside.

Jeg synes is the normal way to express an opinion: I think, I feel, or in my opinion.

• Jeg synes, at ... = I think that ... / I find that ...
• Jeg tænker usually means I’m thinking or I’m considering

So in this sentence, jeg synes is the natural choice because the speaker is giving a judgment about the app.

Yes. At here is a conjunction meaning that. It introduces the clause den nye app er mindre besværlig end den gamle hjemmeside.

So the structure is:

• Jeg synes = I think
• at ... = that ...

Just like in English, it can often be left out in everyday language:

• Jeg synes, at den nye app er ...
• Jeg synes, den nye app er ...

Both are natural.

That is about Danish comma style.

In Danish, there are two accepted comma systems, and one of them places a comma before subordinate clauses such as the one introduced by at. So:

• Jeg synes, at ... is correct
• Jeg synes at ... is also correct in the other comma system

So the comma is not about meaning here; it is mainly about which comma convention is being used.

Because Danish handles definiteness differently depending on whether there is an adjective.

• Without an adjective: appen = the app
• With an adjective: den nye app = the new app

So when you add an adjective like nye, Danish normally uses:

• den/det/de + adjective + noun

Examples:

• appen = the app
• den nye app = the new app
• hjemmesiden = the website
• den gamle hjemmeside = the old website

In a definite noun phrase, Danish adjectives usually take the -e ending.

That is why you get:

• en ny app = a new app
• den nye app = the new app

and

• en gammel hjemmeside = an old website
• den gamle hjemmeside = the old website

So the -e form often appears after den, det, or de.

Because both app and hjemmeside are common gender nouns in Danish.

Their basic forms are:

• en app
• en hjemmeside

Since they take en, the definite adjective form uses den:

• den nye app
• den gamle hjemmeside

If the noun were neuter and took et, you would use det instead.

Mindre besværlig means less troublesome / less cumbersome. It is not the same as besværligere.

• mindre besværlig = less troublesome
• mere besværlig or sometimes besværligere = more troublesome

So in this sentence, mindre is there because the comparison is less, not more.

Also, with longer adjectives, Danish often uses mere/mindre rather than only adding an ending.

Besværlig means something like:

• troublesome
• awkward
• inconvenient
• a hassle to use

In this sentence, it suggests that the new app is easier to deal with or use than the old website. It is about inconvenience or effort, not necessarily difficulty in a technical sense.

End means than in comparisons.

So:

• mindre besværlig end den gamle hjemmeside = less troublesome than the old website

It is the standard word used after comparatives in Danish:

• større end = bigger than
• bedre end = better than
• mindre end = less than

App is a common loanword in modern Danish, and it is widely used in everyday speech and writing.

It behaves like a normal Danish noun:

• en app
• appen
• den nye app

So even though it comes from English, it fits into Danish grammar just like other nouns.

The basic word order here stays:

• den nye app er mindre besværlig ...

That is normal for a subordinate clause introduced by at.

A useful extra point is that if you add a sentence adverb like ikke, it usually comes before the finite verb in a subordinate clause:

• Jeg synes, at den nye app ikke er besværlig.

Compare that with a main clause:

• Den nye app er ikke besværlig.

So the sentence you have is straightforward, but it is a good example of how subordinate clauses work in Danish.