If you are approaching the string with the pick and touching the string just before you strike, then the pick is actually stopping the string from vibrating just before striking the next note.
You may even hear a buzzing sound as you approach the string with the pick. Instead of approaching the string slowly, you should keep the pick clear of the string until you are ready to strike the string right on the beat. So you strike the string quickly and just on the beat, and otherwise keep the pick clear of the vibrating string.
Finger Placement & Holding Strength
Another cause for separated notes may be in the way you are holding the note G on the third fret.
You should be holding the note with your fingertip placed just behind the fret wire.
As your finger placement falls further behind the fret wire, the note becomes increasingly harder to hold down and requires more pressure. If your third finger is well placed, be sure to press down hard enough and with continuous pressure.
If you relax your third finger at anytime while you are playing the first practice measure, the note G will stop sounding, or it will sound muffled, and it may also make a buzzing sound.
As you play the measure again, concentrate on these details and work to attain a legato sound.
Practice Measure 2
The second practice measure displays a different rhythm. In this measure, beat number three is a half note, which rings for two beats. This notes starts sounding on beat number 3, rings through beat number 4, and stops at the moment beat number 1 is played in the next measure.
Play practice measure 2 and count the beat numbers out loud as you play. Remember to count out beat number 4 even though the half note will ring through the beat and you will not strike the string. The sound quality should be smooth and connected as there are no rest in this measure.
When you played measure number 2, did the pattern of rhythm remind you of a song or piece of music you have heard ? If you play the measure twice, the song Jingle Bells may come to mind.
If you have ever been to a sporting event, you may have heard this rhythm, as the crowd stomps their feet to the first two quarter notes and then claps their hands on the half note.
There are many other places where this rhythmic patter can be found, and it is a credit to you as a musician to use your innate ability to recognize patterns of rhythm like this one.