Ben seni görünce seviniyorum.

A word-by-word breakdown is:

• Ben = I
• seni = you as the direct object
• görünce = when/once/as soon as (I) see
• seviniyorum = I become happy / I feel happy / I rejoice

So the structure is roughly:

I + you + when seeing + I am happy

Natural English usually rearranges that as I’m happy when I see you.

Yes. The ending -yorum already shows first person singular, so ben is often unnecessary.

• Seviniyorum = I am happy / I get happy
• Ben seviniyorum = I am happy, with extra emphasis on I

In this sentence, ben may be there for emphasis, contrast, or just because the speaker chose to say it. In everyday Turkish, you could very naturally say:

Seni görünce seviniyorum.

Because sen means you as the subject, but here you is the thing being seen, so it is the direct object.

Turkish marks a definite/specific direct object with the accusative ending.
For sen, that form is:

• sen = you
• seni = you as the object

So:

• Ben seni görüyorum = I see you
• Sen beni görüyorsun = You see me

In your sentence, the speaker is seeing you, so seni is correct.

Görünce comes from the verb görmek (to see) plus the converb/adverbial suffix -ınca / -ince / -unca / -ünce.

Here:

• gör- = see
• -ünce = when / once / as soon as

So görünce means when (someone) sees or upon seeing.

In this sentence, it means:

when I see you

This suffix often connects two actions:

• Eve gelince ararım. = When I get home, I’ll call.
• Seni görünce seviniyorum. = When I see you, I feel happy.

Because görünce is a non-finite verb form. It does not usually take person endings the way a main verb does.

The subject is understood from context, and very often it is the same subject as the main verb.

So in:

Ben seni görünce seviniyorum

the main verb seviniyorum clearly means I am happy, so görünce is naturally understood as when I see.

In other words:

• görünce = when seeing / when [someone] sees
• context tells you that the subject is I

Because of vowel harmony.

The suffix has several forms:

• -ınca
• -ince
• -unca
• -ünce

The form depends on the last vowel of the verb stem. In gör-, the last vowel is ö, which is a front rounded vowel, so the matching form is -ünce.

That gives:

• gör + ünce → görünce

Compare:

• bakınca = when looking
• gelince = when coming
• bulunca = when finding
• görünce = when seeing

Grammatically, seviniyorum is in the present continuous/progressive:

• sevin- = be glad / rejoice
• -iyor = present continuous
• -um = I

So literally it is something like I am becoming glad / I am feeling happy.

But in Turkish, the -iyor form is used more broadly than English am ... -ing. It can express:

• something happening right now
• a repeated reaction
• a present general feeling

So in this sentence, a natural English translation is often:

• I’m happy when I see you
• I get happy when I see you
• Seeing you makes me happy

All can fit depending on context.

It is related in form, but it is a different verb with a different meaning.

• sevmek = to love / to like
• sevinmek = to be glad / to rejoice / to become happy

So:

• Seni seviyorum = I love you
• Seni görünce seviniyorum = I feel happy when I see you

That is an important difference.
Seviniyorum does not mean I love here.

It can often be understood as whenever I see you, especially in a general statement.

Turkish -ince forms can mean:

• when
• once
• as soon as
• sometimes whenever, depending on context

So this sentence could describe a regular emotional reaction:

I’m happy whenever I see you.

If the speaker is talking about a specific occasion, it could also mean:

I get happy when I see you.

Context decides how broad or specific it is.

Yes. Turkish word order is flexible, although the neutral order here is very natural.

Your sentence:

Ben seni görünce seviniyorum

Possible variations include:

• Seni görünce seviniyorum.
• Seni görünce ben seviniyorum.
• Seviniyorum seni görünce.
  less neutral, more marked/emphatic

Turkish often puts the most important or focused information in different places for emphasis. But the basic meaning stays the same.

Yes, you could, and the meaning is very similar: I’m happy when I see you.

But there is a small style difference:

• görünce is shorter and very common in everyday speech
• gördüğümde is a more explicitly personal form, literally something like when I see / at the time that I see

So both are possible, but seni görünce seviniyorum sounds very natural and straightforward.

Yes.

The most common omission is ben, because the verb already shows the subject:

• Ben seni görünce seviniyorum.
• Seni görünce seviniyorum.

Both mean the same thing.

You would usually keep seni, because it tells us who is being seen. If that is already obvious from context, Turkish can sometimes omit it too, but normally it stays:

• Seni görünce seviniyorum. = very natural
• Görünce seviniyorum. = possible only if what/who I see is already understood from context