* Dollar index back below 90; euro above $1.22 * Traders in wait-and-see mode on inflation concerns * Turkish lira unfazed by latest central banker dismissal * Graphic: World FX rates tmsnrt.rs/2RBWI5E LONDON, May 25 (Reuters) - The dollar hit 4-1/2 month lows against a basket of peers on Tuesday, as softer-than-expected U.S. data and insistence from Federal Reserve officials that policy would stay pat allayed investor fears about inflation forcing interest rates higher. The British pound rose, inching back toward the three-month high it reached at the end of last week. The Turkish lira edged lower, largely unfazed by the removal of one of the central bank’s four deputy governors. Investors are heavily short dollars in the belief that low U.S. rates will drive cash abroad as the world recovers from the pandemic. They have become leery of adding to positions after an April leap in inflation cast doubt on the policy outlook, but seemed to find reassurance in data and Fed remarks overnight. The dollar index softened 0.3% to 89.533 in Europe, adding to its 0.2% overnight loss to take it to its lowest since Jan. 7. The euro crossed last week’s four-month top at $1.2245 to hit $1.2262, up 0.3% on the day. “They (Fed speakers) are maintaining the transitory inflation narrative which is pushing back any notion of tapering, which is keeping U.S. nominal yields on the defensive but breakevens are moving higher so real yields are moving further into negative territory and that’s maintaining the weaker dollar narrative,” said Jeremy Stretch, head of G10 FX strategy at CIBC. The U.S. national activity index reading of 0.24 against expectations above 1, as well as dovish comments from Fed speakers, provided some backing for the view that any policy tightening is not happening any time soon. “I think there will come a time when we can talk more about changing the parameters of monetary policy, I don’t think we should do it when we’re still in the pandemic,” Federal Reserve Bank of St. Louis President James Bullard said overnight. The yield on benchmark 10-year Treasuries hovered at 1.5961%, just above a two-week low, and the dollar also eased on the Australian and New Zealand dollars and the yen. The yen was last at 108.735 per dollar while the Aussie and kiwi drifted in the middle of ranges that have held them since April. The Aussie bought $0.7758 and the kiwi $0.7219. Sterling, which has run up about 1.2% over the past three weeks while other majors have steadied or even slipped, was 0.2% higher at $1.4181. The lira eased 0.2% to 8.4070 per dollar. Turkey removed Oguzhan Ozbas from his post as central bank deputy governor, replacing him with Semih Tumen, an adviser to President Tayyip Erdogan. In March, Erdogan sacked central bank chief Naci Agbal, two days after a sharp interest rate hike, and also dismissed another deputy governor later the same month. COMES A TIME Traders have a laser-focus on inflation, the policy response to it and any data or remark that could shed light on either since huge bets on stocks, bonds and currencies are all predicated on the assumption that low rates are here to stay for a while. “Looking ahead, the direction of travel seems to be for further USD weakness in the near-term ahead of the U.S. data and the Fed speakers later today,” said Valentin Marinov, head of G10 FX research at Credit Agricole. “Further out, the risk to any dollar short view is that the upcoming Core PCE data out of the U.S. could revive investors’ inflation fears and support the dollar.” Ahead on Tuesday is a German business survey, a series of U.S. housing updates and a handful of policymaker speeches in Europe, Britain and the United States, which will all be parsed for a reading on inflation. On Thursday, crucial U.S. core consumer price data is due, while in the Antipodes the focus will also be on the Reserve Bank of New Zealand’s Wednesday policy statement, with a risk that it may begin to project rate rises. Cryptocurrencies climbed in Europe after making their best attempt at a bounce yet on Monday from last week’s lows. Bitcoin rose 0.2% to $38,889. Ether gained 0.2% to 2,655, following a 26% rebound on Monday. Reporting by Ritvik Carvalho; additional reporting by Tom Westbrook in Singapore; editing by Jason Neely Our Standards: The Thomson Reuters Trust Principles.