Nobody has mentioned the j-drama Last Friends yet. 

The guy is extremely jealous and possessive, and yet she keeps going back to him. 

Yes, this is an extreme case of jealousy where it has gone too far. That's probably not what most people are looking for, but it is the dark and ugly side of "loving" someone so much that you need to control their every move and start panicking as soon as they're five minutes late, thinking they're cheating on you. It happens in real life, sometimes. That's why I don't like to see it in dramas...

I haven't completed Last Friends yet so I'm not sure I know how I feel about recommending it, but it's a good drama overall if you want something dark, sprinkled with some sweet and heartwarming moments - AND it has that jealousy thing.
High School - Love On. It's subtle and sweet, and the jealousy isn't a raging, angry one. But the guy falls first and tells the girl stay close, and he gets pretty jealous often and has no problem admitting it.
Jealousy Incarnate, literally in the name. 

Watch Full House Thai! Its the best version and you'll LOVE it!
Yeahhhh Full House Thai is wayyyyyy better than the Korean one
I just finished the Thai Full House and it is excellent.  
I'm assuming the jealousy is on the lead guy and not the second lead right? If the second lead is included then add Mirror of the Witch to the list. 

This doesn't happen until later on, but Weightlifting Fairy Kim Bok Jo 
Lucky Romance 
hmm... I think this happens in High School King of Savvy as well? 
Beautiful Gong Shim he gets jealous.  It's more cutesy though, more of a romantic comedy, kinda emphasizing the comedy.  It's really cute, two oddballs fall for each other... Plus the male lead is kinda awesome.  Girl is funny and odd too.  
The male lead gets jealous in Heart to Heart as well.  Nothing over the top though.  
Reply 1997
The Master's Sun
High end crush
Cruel Romance
Marriage not Dating
Empress Ki
Marry Him If You Dare
Surplus Princess
Witch Yoo Hee
The Producers
The Sound of the Desert
Imperial Doctress
Scarlet Heart
Scarlet Heart 2
Secret Garden
Beautiful Gong Shim
Birth of a Beauty
Cheese in the trap
May Queen
Bad Guy
The Equator man
You can try Unstoppable High Kick but you have to wait till the middle
I would say recently, Strong Woman Do Bongsoon
Emergency Couple
Moon Lovers: Scarlet Heart Ryeo
Empress Ki
Boys Over Flowers
I Hear Your Voice
Kill Me, Heal Me
high school - love on

surplus princess

cunning single lady