Ta przeglądarka działa wolno, ale wyszukiwarka znajduje dobre linki.

Because ta przeglądarka is the subject of the sentence: this browser.

• ta = feminine singular nominative
• przeglądarka = feminine singular nominative

In Polish, the subject normally appears in the nominative case.
You would use tę przeglądarkę if browser were a direct object, for example:

• Widzę tę przeglądarkę. = I see this browser.

So here, ta przeglądarka działa wolno means this browser works/runs slowly.

They look similar because they are built in a similar way and are both feminine nouns.

• przeglądać = to browse, to look through

• przeglądarka = browser

• wyszukiwać = to search for
• wyszukiwarka = search engine

The ending -arka often appears in nouns naming a tool, device, or thing that performs some function. So to a learner, it can be useful to notice that both words are related in form, even though they mean different things.

Yes. Działa is from działać, which means to work, to function, or to operate.

So:

• Ta przeglądarka działa wolno
  = This browser works/runs slowly

In technology contexts, Polish often uses działać the way English uses work or run:

• Program działa dobrze. = The program works well.
• Internet nie działa. = The internet isn’t working.

Because wolno is an adverb, and it modifies the verb działa.

• wolny / wolna / wolne = slow or free, depending on context, as an adjective
• wolno = slowly, as an adverb

Here we are describing how the browser works:

• działa wolno = works slowly

Compare:

• To jest wolna przeglądarka. = This is a slow browser.
  Here wolna is an adjective describing the noun przeglądarka.

A useful warning: wolno can also mean it is allowed in other sentences, so context matters.

Znajduje is present tense, 3rd person singular, from znajdować / znaleźć.

Here it means something like:

• finds
• is finding
• tends to find

In a sentence like this, the present tense usually expresses a general or habitual fact:

• wyszukiwarka znajduje dobre linki
  = the search engine finds good links

Polish present tense often covers both simple present and present progressive, depending on context.

Because dobre linki is the direct object of znajduje, and for this type of noun, the accusative plural has the same form as the nominative plural.

• singular: link
• plural: linki

Since linki is a non-masculine-personal plural noun, the pattern is:

• nominative plural: dobre linki
• accusative plural: dobre linki

So:

• Wyszukiwarka znajduje dobre linki.
  = The search engine finds good links.

By contrast, dobrych linków is a different case form, not the one needed here.

Polish often does not need subject pronouns, because the verb ending already shows the person and number.

However, in this sentence, the nouns are stated explicitly because the speaker is contrasting two things:

• Ta przeglądarka działa wolno
• ale wyszukiwarka znajduje dobre linki

So instead of saying it, Polish just names the things directly: the browser and the search engine.

If the context were already clear, Polish might omit a pronoun where English would use it.

Because in Polish, ale = but, and it normally introduces a new clause. A comma is standard before ale.

So:

• Ta przeglądarka działa wolno, ale wyszukiwarka znajduje dobre linki.

This is very natural and correct Polish punctuation.

Yes, Polish word order is more flexible than English word order, because case endings carry a lot of grammatical information.

The given sentence is neutral and natural:

• Ta przeglądarka działa wolno, ale wyszukiwarka znajduje dobre linki.

But other orders are possible for emphasis, for example:

• Wolno działa ta przeglądarka, ale dobre linki znajduje wyszukiwarka.

That sounds more marked or contrastive. A learner should usually stick with the original order unless there is a reason to shift emphasis.

Polish has no articles like English a/an/the.

So:

• przeglądarka can mean a browser or the browser
• wyszukiwarka can mean a search engine or the search engine

The exact meaning depends on context. In this sentence:

• ta przeglądarka clearly means this browser
• wyszukiwarka is understood from context as the search engine or a search engine

Polish uses things like word order, context, and words such as ten / ta / to to express definiteness when needed.

Yes—it's a borrowed word used in Polish, especially in internet and tech contexts.

• singular: link
• plural: linki

Polish often adapts borrowed words to Polish grammar, so even though link originally comes from English, it behaves like a normal Polish noun:

• link
• linku
• linki
• linków

So dobre linki is perfectly normal Polish.

It could. Polish often leaves out a demonstrative like ta unless it is needed.

So these are both possible:

• Ta przeglądarka działa wolno, ale wyszukiwarka znajduje dobre linki.
• Ta przeglądarka działa wolno, ale ta wyszukiwarka znajduje dobre linki.

The second version adds extra emphasis, something like but this search engine finds good links.
Without ta, the sentence sounds more natural and less repetitive.