It is not just buses that come in threes. The government has suffered a trio of reversals in recent days. First, the big revolt by Tory MPs against their ministers' attempts to accommodate the European court's ruling that prisoners should have the vote. Then, the education secretary was told to sit on the naughty step by the British high court which declared that Michael Gove was guilty of an "abuse of power" when he scrapped the school building programme without consultation. Third, the government beat a retreat over the privatisation of woodlands managed by the Forestry Commission. That ill-conceived wheeze has been stalled while ministers "re-examine the criteria". A monster backbench rebellion, a stinging rebuke from a senior judge and a messy U-turn – all in the space of just one week.

None of these events is anywhere near life-threatening for the government, but they do put a useful spotlight on some of its flaws.

Weakness number one is a cumulative amateurishness. Some of this can be put down to sheer inexperience. Thanks to the novelty of coalition and the intensity of its activity, commentators have tended to be distracted from the simple but important fact that we have a government of novices. This was an inevitable result of a coalition between one party that had been out of power for 13 years and another party that had not held a red box in more than 60 years. Before May, none of the Lib Dem members of the cabinet had been so much as parliamentary under-secretary for paperclips. Of the Tories around the top table, only Ken Clarke, William Hague and Sir George Young had sat there before.

This inexperience was largely masked for a while, not least because David Cameron is highly accomplished at the frontman aspect of being prime minister. I believe he went to a special academy near Slough where they train their students to look as if they were born to rule. But that inexperience is now becoming manifest in policy blunders. Third-order initiatives, such as the plan to sell-off the Forestry Commission, are launched into the public domain in an ill-thought-out way, come under attack from all sides when this government is doing quite enough that is unpopular to be going on with, and then have to be ditched. "Bloody silly idea from the off," is how one Tory minister characterises flogging off ancient woodlands.

Around the prime minister, there is angst and debate about how to strengthen the Number 10 operation in order to prevent ministers from blundering into avoidable traps and scoring own goals. It is argued that David Cameron needs to improve his early-warning radar to spot and defuse incoming trouble before there is a public debacle. Some suggest that the prime minister needs to beef up his policy unit and make it more aggressive about intervening to stop departments lurching into calamity. Others contend that the last thing they need in Number 10 or at the Cabinet Office is to recruit any more pointy heads. One senior Conservative complains that the real problem is that: "We have a government of old Etonian intellectuals and policy wonks. They think of the policy, not the politics."

Inexperience is magnified as a handicap when it is compounded with two further weaknesses: impetuosity and ideology untempered by common sense. Before the formation of this government, it was often said, especially by David Cameron, that coalition would be a disastrous form of government because it would be feeble and indecisive. As it turns out, that is one criticism that cannot be fairly directed at them. For good or ill, across a wide range of areas, from welfare to the constitution, the coalition is shaking things up. David Cameron has been so determined, as has Nick Clegg, to prove how bold and radical they can be that they have gone to the other extreme.

I can still find no one at senior levels of government who is truly and wholly convinced that it is a sensible idea to add a massive upheaval of the National Health Service to their groaning table of challenges. Not, at the very least, without first piloting the Lansley plan to hand over £80bn of spending power to GPs. If you could administer a truth serum to Mr Cameron and Mr Clegg, I bet both would confess that they are not fully confident that it will work out. The Lib Dem leader has been heard to say to colleagues that, if this goes wrong, the Lansley plan all by itself could lose them the next election. I strongly suspect that Mr Cameron signed off on it not because he is totally convinced that it will work, but because he feared he would look wussy if he put on the brakes.

Michael Gove took power at education determined to make a reforming splash from the start and convinced that Labour's school rebuilding programme was wasteful and badly managed. He was not all wrong about that. His error – the mistake which exposed him to legal challenge – was to swing his axe without discrimination or consultation rather than examine each project on its merits.

I expect the education secretary will have plenty of company in the dock in the months ahead. There will be lots more examples of ministers landing themselves in the merde because they have been too eager to prove their machismo as cutters and too splapdash about implementing the spending squeeze. These are flaws that can be addressed. Experience of governing ought to come from, well, experience of governing. That should also temper reckless urges to act first and ask questions later. Ministers who don't learn to do better can always be sacked.

That leaves the more structural weaknesses of the coalition. One is that it remains very much a top-down project. David Cameron and Nick Clegg have sustained a decent relationship. It has its ups and downs, but it is a more civil and trusting partnership than either probably thought possible when they first got hitched. Most Tory and Lib Dem ministers are still rubbing along with each other all right, but that amity has not extended more deeply down into the ranks. This is a coalition joined at the head, but not at the hip. Resentment and suspicion that too much is being conceded to the other side are to be found among both Tories and Lib Dems. Many of the latter think the Conservatives have let off their banking friends. The tension erupted on the Tory side when Conservative backbenchers revolted in great numbers over votes for prisoners. Tory MPs were never going to be happy about that and even less so when the demand comes from the European Court of Human Rights.

This problem for David Cameron is not a result of coalition. The same dilemma would face a purely Conservative government. But the vehemence of Tory feeling on this issue has been swollen by their disgruntlement about the compromises of coalition; there was a strong impression during the Commons debate that Conservative MPs were letting off steam. After the vote, I caught up with one senior Tory who was in a state of some euphoria. He said joyously: "At last we could vote for something we really believed in."

In David Cameron and Nick Clegg, the coalition has leading men who made their names and won their positions because they are extremely fluent performers. And yet the coalition has a big problem communicating with both their own parties and the public. This week, beginning with his article in today's Observer, David Cameron will be attempting to rescue the "big society", an operation he would not be mounting if it were not at serious risk from terminal ridicule. Nick Clegg has had some success lately overcoming the recalcitrance of the Home Office and getting the government committed to repealing much of New Labour's authoritarianism. But the Lib Dem leader still struggles to make his "freedom agenda" sing with the voters.

Why are these two skilled communicators failing to convince the public that there is more to this government than just spending cuts? Well, it is obviously partly because spending cuts are such a big element of the story. But it is also because their coalition has tended to be a rather incestuous form of government. Tory and Lib Dem ministers have too often assumed that once they've wrangled behind closed doors and struck a deal between themselves, then a problem is solved.

There is a tendency to think that if they can make a policy palatable to each other, then it must be fine. The Lib Dems believed they had struck a decent bargain with the Tories on tuition fees and arrived at a superior policy – they really did. Their mistake was to forget about the voters and not think more about how such a blatantly broken promise would go down with the public.

In theory, two parties, representing as they do different political traditions, parts of the country and electoral constituencies, ought to be more attuned to and responsive to the voters. It is of concern to thoughtful people at the top of government that it has not so far turned out that way. The amount of time spent negotiating with each other has made the coalition inward-facing when it needs to be much more public-facing. They forget at their peril that the most important negotiation in politics is always with the voters.