David Cameron is facing huge embarrassment after details of intimate texts he exchanged with the former News International chief Rebekah Brooks emerged on Saturday.

The existence of the messages has been at the centre of a row between Labour MP Chris Bryant and Cameron for several weeks and the contents will be a thorn in the prime minister's side. In one message, Cameron thanks Brooks for letting him ride one of her family's horses, saying it was "fast, unpredictable and hard to control but fun", it is claimed.

In another, it is reported that Brooks, who is standing trial next year on charges of phone hacking and conspiracy to pervert the course of justice, tells the Tory leader that she felt so emotional listening to his conference speech she "cried twice", adding: "Will love 'working together'."

The messages are part of a cache of texts and emails between Downing Street and Brooks handed over to Lord Justice Leveson's inquiry into media ethics by Downing Street and Brooks. In one, Cameron writes: "The horse CB [Charlie Brooks] put me on. Fast, unpredictable and hard to control but fun." He signed off "DC".

Bryant tweeted: "These new texts are the tip of an iceberg." He also confirmed he has written to Leveson demanding that he publish all the texts and emails between the pair. Separately, the Observer understands there may have been as many as 150 messages exchanged between the two over a number of months, many of which would prove to be a considerable embarrassment to the government.

Only a small number of the messages have so far been made public, leading to claims by Bryant, a former Europe minister, of a cover-up.

A furious Cameron rejected a demand in the House of Commons by Bryant two weeks ago to disclose all the texts and emails exchanged with Brooks, whose husband, the racehorse trainer Charlie Brooks, has been friends with Cameron since they both attended Eton. The prime minister has refused to address Bryant's complaints on the basis that he broke an embargo on details disclosed to the Leveson inquiry and is said to have refused to apologise.

But in a recent exchange, Bryant said: "When the truth comes out, you won't be smiling."

On Saturday it was also said that Bryant claims to have had contact with a No 10 "mole" involved in compiling Cameron's texts and emails, whom the Labour MP says has confirmed their "salacious" nature. Both texts revealed on Saturday are said to have been sent in October 2009, shortly after Brooks left her job as editor of the Sun and became chief executive of News International, which owns the paper.

They shed further light on the extraordinarily close relationship between Cameron and Brooks, both members of the "Chipping Norton set" based around their Oxfordshire homes.Brooks told the Leveson inquiry earlier this year that Cameron signed some of his missives to her "LOL" – until she told him it meant "Laugh Out Loud", not "Lots Of Love".The disclosures are bound to lead to further Labour party demands that all messages between the Brooks and Cameron are made public so the true nature of their relationship can be seen.

Cameron and Brooks met 22 times in a six-year period, an average of once every three months. Brooks said that at the height of the phone-hacking scandal he sent her a morale-boosting "keep your head up" message, and expressed his regret he could not be more loyal to her in public.