Clarke hopes the focus will now be on Albion's success this season in sustaining a top-half-of-the-table finish in the Barclays Premier League rather than the Nigeria international.
Odemwingie has attracted plenty of headlines for a series of attacks against the club via the social networking site and his latest outbursts have been dealt with internally.
Clarke said: "The matter (Odemwingie) has been dealt with internally by the club. I would like to move on from it and with seven games remaining.
"Hopefully we can start talking about West Brom and the good season we've had up to now and hopefully more good things to talk about between now and the end of the season."
But Clarke said: "Everyone has to be open-minded about Twitter, although it's not a concept I'm comfortable with. I don't understand the attraction of faceless media.
"I try to sort things out face-to-face with people and I can understand the tone of questions better when they come from someone in the same room.
"Social media, including emailing, is impersonal but I have to try to understand it better.
"A lot of people obviously love it. I've got kids who use it."
Odemwingie could be part of the Baggies squad to face Arsenal in Saturday's Premier League clash at the Hawthorns.
But he will be without midfielder Youssouf Mulumbu who has to serve a three-game ban for his red card against West Ham after kicking the ball at midfielder Gary O'Neil.
Clarke said: "Again it is an internal disciplinary matter. He will be dealt with. He was very apologetic at the time.
"I think you could just see it was out of sheer frustration at a disappointing performance from us.
"It was completely out of character for him. It is disappointing for him and us because we lose him for three games and he let himself down.
"But he apologised and that was the best thing to do."