Praise be to Allah.
We have to differentiate here between two important matters with regard to this issue.
The ruling on this correspondence between young men and young women, and entering into relationships in which they get to know one another and talk to one another. This is something that is haraam. We have drawn attention to this prohibition in several answers which you can read on this site.
It cannot be justified by saying that the intention of one or both parties is to look for a suitable partner to marry. This is looking in an unknown world where it is not possible to verify any of the characteristics that the suitor should be looking for, let alone the lack of the required seriousness when the matter is approached in this way, and the lack of honesty in describing the characteristics of each party, and other matters which anyone who has approached this matter in this way knows.
In the event that this has actually taken place, i.e., where a person has started to get to know someone -- whether that was because he did not know that such relationships are haraam, or because he thought that the good intention (seeking a suitable spouse) made it permissible, or he embarked on it as most people do, without paying attention to whether it is halaal or haraam, or because he was following his whims and desires -- if the person in such a situation asks us whether he can go ahead with this marriage or not, the answer is:
In general we do not advise these kinds of marriages, because the element of certainty with regard to the information acquired about the other party is not present; rather it is information that each party has given about himself or herself, which may lead to infatuation between the two parties before it has been established whether they are suitable for marriage or whether they have the qualities needed for a successful marriage.
But despite that there may be some cases in which going ahead with marriage may be acceptable, despite the fact that the initial steps towards the marriage were wrong, as they came about because of ignorance of the ruling, as we have said, or because of a mistake which was out of character for the individual, and it has been possible to find out the necessary information about each party. In this case it may be acceptable -- in certain cases -- to go ahead with a marriage that started in this manner. The permissibility is further confirmed if the attachment to one another has become so strong that it is difficult for them to be separated. In this case we advise them to repent from this haraam relationship and they should be reminded to adhere to the commands of Allah, then there is nothing wrong with them going ahead with the marriage.
And Allah knows best.