One of the drawbacks of written musical notation is that the rests are often more visually predominant than the notes. One note might be followed by many rest symbols before the next note appears. Players often attach such importance to or are so overwhelmed by the rests that they play the following note(s) out of time.

My suggestion to overcome this is to imagine each note to be full length up to the next one. This will keep all the notes within their correct rhythmic framework. Once this is comfortable, shorten the notes to comply with their particular note values and play the passage in the correct rhythm. The rests will take care of themselves, they are only there to fill in the gaps.